Functorial Signal Representation & Base Structured Categories
Salil B. Samant
Department Of Electrical Engineering INDIAN INSTITUTE OF TECHNOLOGY DELHI
January 2018
Functorial Signal Representation & Base Structured Categories
by
Salil B. Samant
Department of Electrical Engineering
submitted in fulfillment of the requirements of the degree of Doctor of Philosophy to the
INDIAN INSTITUTE OF TECHNOLOGY DELHI
January 2018
To
My Wife for loving me the way I am and agreeing to walk life’s journey with me
My Mother for being first teacher and never forcing personal ambitious desires on her child
Alexander Grothendieck for renunciation of fame, awards, wealth, family and setting an example for the new generation to follow
Beloved Jesus, Koot Hoomi, D.K & Lord Maitreya for showing the inner path of light when all outer lights went out
i
Certificate
This is to certify that the thesis titled Functorial signal representation and Base structured categories, submitted by Salil Samant, to the Indian Institute of Technology, Delhi, for the award of the degree of Doctor of Philosophy, is a bona fide record of the research work done by him under my supervision. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma.
Shiv Dutt Joshi Professor Dept. of Electrical Engineering IIT-Delhi, 110016
Place: New Delhi
Declaration
I declare that this dissertation is the result of my own work representing my original intuitions and ideas in my own words and where others ideas or words have been included, I have adequately cited and referenced the original sources to the best of my knowledge. I also declare that I have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or falisfied any ideas/data/fact/source in my submission. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma.
Salil Samant
Date: 10 January 2018
Acknowledgements I would like to express my deepest gratitude to my supervisor Professor S.D Joshi for his unwavering support and ensuring peaceful environment throughout this work which I believe nobody could have provided. I will forever be thankful to him for giving the opportunity and full support to pursue my interest without any boundaries whatsoever. I thank my review committee members Dr. Monika Aggarwal, Dr. Mashuq un Nabi, Dr. S. Janardhanan Dr. Shankar Prakriya for their inputs in my research. Special thanks to Dr. Monika Aggarwal for her teaching class Sensor array signal processing during my Mtech journey and introducing the reference book Statistical signal processing by S.M Kay in the class. I was fortunate enough to be able to interact with her during my stay at IITD both during my Mtech and PhD journey. Many thanks to Dr. Mashuq un Nabi and Dr. S. Janardhanan for inculcating my interest in systems engineering and their intuitive approach in classes such as optimal control and linear systems theory. I am indebted to both of them for this. I would like to thank Category Theory meeting 2016 organizing team especially Professors Peter Hofstra and Richard Blute, for providing me an opportunity to present my abstract covering the underlying intuition. Finally I would like to thank my family which includes my wife, aai, papa for their constant support.
Salil Samant
i
Preface The thesis “Functorial signal representation and Base structured categories” unfolds the intuition underlying compression or in modern terminology sparse nature of a given signal representation by penetrating into the heart of natural generative cause behind the signal. During my M.tech journey in signal processing almost a decade ago, I was largely influenced in particular by the intuitive visual approach of certain professors especially Prof. Steven M. kay in statistical signal processing theory and Prof. Gilbert Strang in linear algebra. These revered teachers had an extraordinary ability to cut through the ocean of mathematical jargon and make students see what really lies behind the mathematical formulas. Since then I had a deep desire to understand the structure of information carried by signals (such as a music signal or a painting invoking ecstasy) and always thought that we should be able to represent a signal in the way it was conceived or generated. However not knowing how to proceed on this line of thought I dropped the plan of pursuing this further for the time being. By the end of year 2011 my search while working in industry lead me to the book "Generative theory of shape" which confirmed that I am not a lone ranger among such thinkers and Prof. Michael Leyton was among them conceptualizing a generative cause behind concrete expressions of art, shape, music etc. While grasping the wreath group approach (utilizing "Visual group theory" by Prof. Nathan Carter) towards the generative structure of many natural processes, I realized that it is the special case of something called ‘Grothendieck Fibration’ in category theory. With this began the earnest study of category theory since mid 2014. I used to present my intuitive understandings of category theory every morning to my guide and we had a lot of fruitful discussion which sparked his interest back into category theory which he first learnt from his mathematics teacher (late) Prof J.B ii
iii Srivastava. He was very supportive of my desire to explore this new theory in the quest of my original problem. With my guide I was sure of getting peaceful time to do whatever I wanted to pursue. But of-course with freedom comes responsibility. He used to always encourage me saying “Salil intuition is more important than just precision”, whenever I used to feel that I am seriously lacking in mathematical precision which I know I still lack. Moreover as and when I am worried about possibility of erring using pure visual intuitive approach, the following quote by Prof. Grothendieck is the key thing I remind myself often, “Discovery is the privilege of the child: the child who has no fear of being once again wrong, of looking like an idiot, of not being serious, of not doing things like everyone else”. By taking a fresh perspective of category theory on Leyton’s model I was able to feel some deeper significance of whole arrow approach. In fact soon I realized that redundancy in signals can only be understood via relative approach of arrow. Of-course once we are able to precisely explain what redundancy itself is, I feel the reason for compression in particular representations will naturally get revealed. This is one central theme explored in the thesis. Moreover the intuitive approach has led to certain new results within category theory itself which I have termed as ‘Base structured categories’ as well the philosophy of functor acting as a general generative system inherently converting the underlying abstract virtual cause into concrete observable effect in real world as shown in following diagram: Cause
Functor
Effect
In conclusion the thesis is largely influenced by Leyton’s generative theory of shape and Grothendieck’s relative viewpoint or more generally the arrow perspective of category theory and tackles the fundamental question of what is redundancy and why compression precisely occurs in a representation. I hope the reader enjoys reading the way I have enjoyed exploring ! Salil Samant c 2018, Indian Institute of Technology Delhi
Abstract The study of categories (F, C, D) and (F, C, Set) as graphs of a functor leads to a proof of these being fibred on C. Grothendieck construction in the context of an ordinary functor F : C → D is examined through the concept of trivial categorification, R F I ¯ This category using an appropriate functor F : C − →D→ − Cat to construct Cop F. characterizes a functor as an abstract right category action while its dual X oF C R ¯ op characterizes a functor as an abstract left category action. Similarly or ( Cop F) R F U ¯ as categories functor F : C − →D− → Set leads to definition of X oF C and op F C
denoting concrete left and right actions of C respectively. Collectively referred to as ‘base structured categories’, these are proven abstractly isomorphic to the base R ¯∼ category C but concretely isomorphic to each other or (F, C, Set) ∼ = X oF C. = CF These are special instances of fibred categories where the base category is C and fibres are D objects being treated as trivial categories. The perspective of making only base structure explicit through category theory concealing the vertical structure using identity morphisms enables one to combine intuitions of Grothendieck’s relative and Leyton’s generative theory. As explored further, it greatly facilitates the application of functors in certain fundamental applications such as signal representation which hitherto have been treating objects of category D such as Hilbert spaces or Riesz spaces purely in a set theoretic way. Next the foundations of a functorial framework for representing signals are proposed using a functor assisted by base structured categories. By incorporating additional category-theoretic relative and generative perspective alongside the classic set-theoretic measure theory the fundamental concepts of redundancy, compression are formulated in a novel authentic arrow-theoretic way. The existing classic framework representing a signal as a vector of appropriate linear space is shown as a special case of the proposed framework. Further in context of signal-spaces as a categories, various covariant and contravariant forms of L0 and L2 functors using categories of measurable or measure spaces and their opposites involving Boolean and measure
iv
v
algebras along with partial extension are studied. We contribute a novel definition of intra-signal redundancy using general concept of isomorphism arrow in a category covering the translation case and others as special cases. Through category-theory we provide a simple yet precise explanation for the well-known heuristic of lossless differential encoding standards yielding better compressions in image types such as line drawings, iconic image, text etc; as compared to classic lossy representation techniques such as JPEG which choose bases or frames in a global Hilbert space. Finally as an extended exploration of Transformation categories few applications R R ¯ are proposed and ¯ X oF C and of the base structured categories X oF C, C F, F C studied. Classic transformation groupoid X//G is proven as being a base-structured R category G F¯ . Then using permutation action on a finite set, we introduce the notion of a hierarchy of base structured categories [(X2a oF2a B2a )q(X2b oF2b B2b )q...]oF1 B1 that models local and global structures as a special case of composite Grothendieck fibration. Further utilizing the existing notion of transformation double category (X1 oF1 B1 )//2G, we demonstrate that a hierarchy of bases naturally leads one from 2-groups to n-category theory. Finally we prove that every classic Klein geometry is F U the Grothendieck completion (G = X oF H) of F : H − → Man∞ − → Set. This is generalized to propose a set-theoretic definition of a groupoid geometry (G, B) with a principal groupoid G = X o B and geometry space X = G/B; which is essentially F U same as G = X oF B or precisely the completion of F : B − → Man∞ − → Set.
KEYWORDS:
Base structured category, Functor, Grothendieck fibration, Leyton transfer, Functorial signal representation, Redundancy.
c 2018, Indian Institute of Technology Delhi
Contents
Acknowledgements
i
Preface
ii
Abstract
iv
List of figures
xiv
List of tables
xv
Notation and terminology
xvi
1 Introduction
1
1.1
Background and motivation . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Structure and contributions . . . . . . . . . . . . . . . . . . . . . .
8
2 Preliminaries 2.1
2.2
13
Category theory and Grothendieck fibration . . . . . . . . . . . . .
13
2.1.1
Category and functor . . . . . . . . . . . . . . . . . . . . . .
13
2.1.2
Product, pullback and limits . . . . . . . . . . . . . . . . . .
16
Fibred category theory . . . . . . . . . . . . . . . . . . . . . . . . .
22
2.2.1
22
Set-Indexed Sets . . . . . . . . . . . . . . . . . . . . . . . . vi
CONTENTS
2.3
vii
2.2.2
Change of Base . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2.3
Fibration . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.4
Basic Properties of Fibration
. . . . . . . . . . . . . . . . .
27
2.2.5
Cloven and Split Fibration . . . . . . . . . . . . . . . . . . .
29
2.2.6
B-Indexed Category and Grothendieck Construction . . . . .
32
2.2.7
An example of fibration, P : Vect → Fld . . . . . . . . . . .
36
2.2.8
Connection of (F, C, Cat) to fibred category . . . . . . . . .
38
Measure theory and functional analysis . . . . . . . . . . . . . . . .
40
2.3.1
σ-algebra and measure space . . . . . . . . . . . . . . . . . .
40
2.3.2
Boolean algebras and measure algebras . . . . . . . . . . . .
43
2.3.3
Atomic measure spaces and dual atomic Boolean algebras. .
48
2.3.4
Measurable, measure-preserving functions and Boolean homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
On spaces: L0 , L2 , L0 , L2 . . . . . . . . . . . . . . . . . . .
54
2.3.5
3 Base structured categories 3.1
3.2
60
Functor graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
3.1.1
The category (F, C, D) . . . . . . . . . . . . . . . . . . . . .
61
3.1.2
Remarks on relationship of (F, C, D) to Graph(F) . . . . . .
67
Transformation categories . . . . . . . . . . . . . . . . . . . . . . .
67 68
3.2.2
Trivial categorification of D objects . . . . . . . . . . . . . . R R ¯ . . . . . . . ¯ op F Right action induced by a functor: Cop F, C
3.2.3
Left action induced by a functor: X oF C , X oF C . . . . .
72
3.2.1
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69
CONTENTS
3.2.4
viii
Duality between the categories defined as right actions and left actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Functorial signal representation 4.1
4.2
4.3
74 82
Functorial signal model . . . . . . . . . . . . . . . . . . . . . . . . .
83
4.1.1
Underlying intuition . . . . . . . . . . . . . . . . . . . . . .
86
4.1.2
Case of basic measurable structure . . . . . . . . . . . . . .
87
4.1.3
Trivial arrows versus non-trivial arrows . . . . . . . . . . . .
88
4.1.4
Collection of independent elements versus arrow based generalized elements and differentials. . . . . . . . . . . . . . . . . .
89
4.1.5
Signal space as a signal matched category . . . . . . . . . .
90
4.1.6
Invoking functorial L2 construction for special class of inversemeasure preserving maps on measure spaces . . . . . . . . .
91
4.1.7
Change of base and Grothendieck’s relative point of view . .
92
4.1.8
Arrow-theoretic understanding of redundancy and compression
93
4.1.9
Equivalent formulation using Meas2 . . . . . . . . . . . . .
95
4.1.10 Dealing partial isomorphisms, practical observation errors and general limitations. . . . . . . . . . . . . . . . . . . . . . . .
96
4.1.11 A prototypical example beneath the design of Portable network graphics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
Redundancy and examples . . . . . . . . . . . . . . . . . . . . . . .
103
4.2.1
Special cases of redundancies . . . . . . . . . . . . . . . . .
105
4.2.2
Examples: groupoid base structures in signals . . . . . . . .
107
4.2.3
A base structured category perspective . . . . . . . . . . . .
109
Categories for L0 , L2 functors . . . . . . . . . . . . . . . . . . . . .
114
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CONTENTS
4.4
4.5
ix
4.3.1
Base or domain categories C, Cop . . . . . . . . . . . . . . .
114
4.3.2
Objects of C, Cop . . . . . . . . . . . . . . . . . . . . . . . .
115
4.3.3
Arrows of C, Cop . . . . . . . . . . . . . . . . . . . . . . . .
117
4.3.4
Codomain categories . . . . . . . . . . . . . . . . . . . . . .
119
On L0 ,L2 functors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120
4.4.1
Results on functors L0 . . . . . . . . . . . . . . . . . . . . .
120
4.4.2
Results on functors L2 . . . . . . . . . . . . . . . . . . . . .
125
4.4.3
Generalization to partial categories . . . . . . . . . . . . . .
130
4.4.4
The L0 functor extended to partial category . . . . . . . . .
133
4.4.5
The case of linear Borel measurable function . . . . . . . . .
134
Functorial signal space properties . . . . . . . . . . . . . . . . . . .
135
4.5.1
Linear structure . . . . . . . . . . . . . . . . . . . . . . . . .
136
4.5.2
Partial order and lattice structure . . . . . . . . . . . . . . .
137
4.5.3
Multiplicative structure
138
. . . . . . . . . . . . . . . . . . . .
5 Some applications of transformation categories 5.1
5.2
140
From group action to category action . . . . . . . . . . . . . . . . .
140
5.1.1
Sets as categories . . . . . . . . . . . . . . . . . . . . . . . .
142
Base structured categories in symmetry . . . . . . . . . . . . . . . .
143
5.2.1
Global symmetry using a single object base category . . . .
144
5.2.2
Local and global symmetries using a multi-object base category 145
5.3
Hierarchy of symmetry I: Leyton’s generative theory . . . . . . . . .
147
5.4
Hierarchy of symmetry II: Base structured categorical models . . .
152
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CONTENTS
5.5
x
5.4.1
Hierarchy of symmetries . . . . . . . . . . . . . . . . . . . .
152
5.4.2
Using hierarchy of Groups [(X oF2 (G3 × G3 )] oF1 G2 . . .
153
5.4.3
Using hierarchy of groupoids [(X2a oF2a G3 )q(X2b oF2b G3 )]oF1 G2
155
5.4.4
Connection of hierarchy of base structured categories with standard composite of Grothendieck fibrations . . . . . . . . . .
157
Hierarchy of structures: 2-groups to n-category theory . . . . . . . .
158
5.5.1
The naturality square corresponding to inner 2-group action
161
5.5.2
The naturality square at a node of outer naturality square .
162
5.5.3
The naturality square corresponding to outer 2G action . . .
163
5.6
Geometries as base structured categories X oF C . . . . . . . . . .
164
5.7
Multi-object species of structures: Heuristic discussion . . . . . . .
168
6 Conclusion and extension
175
References
177
A Some results on partial categories
183
A.1 Partial Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . .
183
List of papers based on thesis
185
Brief biodata of author
186
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List of Figures
1.1
Functor as a system connecting abstract cause with concrete effect.
1
1.2
Functor as a system connecting sheet melodies with concrete signals
2
1.3
Functor as a system connecting real objects with concrete signal . .
3
1.4
Functor as a system connecting linguistic with concrete signals . . .
3
2.1
Universal property of a product . . . . . . . . . . . . . . . . . . . .
17
2.2
Heuristic of a Pullback in FinSet . . . . . . . . . . . . . . . . . . .
19
2.3
Inverse image of a function as a pullback . . . . . . . . . . . . . . .
19
2.4
A functor D and a cone with vertex N . . . . . . . . . . . . . . . .
20
2.5
Limit of a diagram D
. . . . . . . . . . . . . . . . . . . . . . . . .
21
2.6
Change of base for set-indexed sets. . . . . . . . . . . . . . . . . . .
24
2.7
Commutative squares in the mixed category . . . . . . . . . . . . .
27
2.8
Cartesian lift as a pullback in mixed category . . . . . . . . . . . .
27
2.9
Pullback functor from EJ to EI determined by cartesian lifts . . . .
30
2.10 Cartesian lift of identity in a cloven fibration . . . . . . . . . . . . .
31
2.11 Cartesian lift of composite arrow in a cloven fibration . . . . . . . .
31
2.12 Coherence conditions for u : I → J . . . . . . . . . . . . . . . . . .
32
2.13 Coherence conditions for I →u J →v K →w L . . . . . . . . . . . .
32
2.14 Proof for the cartesian lifts in fibration formed using a strict functor Ψ
35
xi
LIST OF FIGURES
xii
2.15 Classic fibration definition in the case of strict functor Ψ . . . . . .
36
2.16 Category of vector spaces as a fibration . . . . . . . . . . . . . . . .
38
2.17 Category (F, C, Cat) characterizing F
. . . . . . . . . . . . . . . .
38
2.18 Expansion of the node (X, F X) . . . . . . . . . . . . . . . . . . . .
39
2.19 Schematic of the category with (X, F X),(Y, F Y ) expanded. . . . . R 2.20 C F¯ fibred on C . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39 40
3.1
(F, C, D) with trivial categories as fibres, on C
3.2
Pullback functor f ∗ corresponding to f in fibration P : (F, C, D) → C
65
3.3
68
3.5
Trivial categorification . . . . . . . . . . . . . . . . . . . . . . . . . R ¯ fibred on Cop . . . . . . . . . . . . . . . . . . . . . . . . . . F Cop R ¯ fibred on Cop . . . . . . . . . . . . . . . . . . . . . . . . . . . F C
3.6
X oF C fibred on C . . . . . . . . . . . . . . . . . . . . . . . . . .
73
3.4
. . . . . . . . . . .
X oF C fibred on C . . . . . . . . . . . . . . . . . . . . . R ¯ fibred on C ∼ 3.8 C F = Cop . . . . . . . . . . . . . . . . . . . R ¯ fibred on C ∼ 3.9 C F = Cop . . . . . . . . . . . . . . . . . . . R ¯ 3.10 Concrete isomorphisms between (F, C, Set), X oF C, C F 3.7
64
71 72
. . . . . .
74
. . . . . .
76
. . . . . .
77
. . . . . .
80
4.1
Functorial signal representation . . . . . . . . . . . . . . . . . . . .
84
4.2
Signal space as a subcategory of Hilb or Riesz . . . . . . . . . . .
84
4.3
Illustration of global and local time signals . . . . . . . . . . . . . .
86
4.4
Prototype differential coding example underlying PNG . . . . . . .
99
4.5
Functorial signal representation model for prototype . . . . . . . . .
100
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LIST OF FIGURES
xiii
4.6
Functorial signal representation model for sequence . . . . . . . . .
102
4.7
Redundancy between local signal . . . . . . . . . . . . . . . . . . .
104
4.8
A simple iconic image of umbrella [id] . . . . . . . . . . . . . . . . .
108
4.9
Surface 3D plot of filtered umbrella image . . . . . . . . . . . . . .
109
4.10 Sample photographic cameraman image [cam] . . . . . . . . . . . .
109
4.11 2D plot of filtered cameraman . . . . . . . . . . . . . . . . . . . . .
110
4.12 Surface 3D plot of filtered cameraman
. . . . . . . . . . . . . . . .
110
4.13 BBC audio signal with translation transfer . . . . . . . . . . . . . .
111
4.14 Transfered melodies in sample audio superimposed. . . . . . . . . .
111
5.1
symmetry of a set using group as a base category. . . . . . . . . . .
145
5.2
Symmetry of a set using groupoid as a base category. . . . . . . . .
146
5.3
2-level symmetry illustration . . . . . . . . . . . . . . . . . . . . . .
148
5.4
Symmetry of prototype X1 captured by D3 . . . . . . . . . . . . . .
148
5.5
Leyton’s generative model using groups . . . . . . . . . . . . . . . .
149
5.6
Leyton’s generative model using groupoids . . . . . . . . . . . . . .
151
5.7
Leyton’s generative model using base structured categories . . . . .
151
5.8
Base structured categorical model of Leyton’s generative theory . .
152
5.9
Inner symmetry using groups . . . . . . . . . . . . . . . . . . . . .
153
5.10 Outer symmetry using groups . . . . . . . . . . . . . . . . . . . . .
154
5.11 Inner symmetries using groupoids . . . . . . . . . . . . . . . . . . .
155
5.12 Outer symmetry using groupoids . . . . . . . . . . . . . . . . . . .
156
5.13 Hierarchy of naturality squares leading to 3-,4-morphisms . . . . . .
160
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List of Tables
1.1
Fibred category versus base structured category . . . . . . . . . . .
5
1.2
Category action on objects . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
Category action on underlying sets of objects . . . . . . . . . . . . .
8
4.1
conventional and functorial signal model . . . . . . . . . . . . . . .
85
4.2
Conventional versus proposed signal representation model . . . . . .
87
4.3
Different mathematical expressions . . . . . . . . . . . . . . . . . .
88
4.4
Trivial versus non-trivial arrows . . . . . . . . . . . . . . . . . . . .
89
4.5
Independent versus generalized elements . . . . . . . . . . . . . . .
90
4.6
Function space versus subcategory in Hilb or Riesz . . . . . . . . .
91
4.7
Function space versus category for affine transfomations . . . . . . .
92
4.8
fixed base versus multiple bases . . . . . . . . . . . . . . . . . . . .
93
4.9
Redundancy and compression . . . . . . . . . . . . . . . . . . . . .
95
4.10 Differential coding prototype example values . . . . . . . . . . . . .
102
4.11 Base categories covering various cases of functors . . . . . . . . . .
115
4.12 Objects of various base categories for functors . . . . . . . . . . . .
116
4.13 Arrows of various base categories for functors . . . . . . . . . . . .
119
4.14 Combination of structures in codomain categories . . . . . . . . . .
120
4.15 Summarized duality between covariant and contravariant functors .
131
xiv
LIST OF TABLES
xv
5.1
From Klein geometry to groupoid geometry . . . . . . . . . . . . .
166
5.2
Single-object against multi-object species of structures . . . . . . .
171
5.3
Structure induced in D relative to different bases . . . . . . . . . .
172
5.4
Structure induced in Set relative to different bases . . . . . . . . .
173
5.5
Structure induced in Set relative to different bases . . . . . . . . .
173
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Notation and terminology In thesis, the upright Roman boldface font such as C or Riesz denote categories. In particular, if we want to emphasize the concrete nature of a particular category the notation (C, U ) is used. Functors in general are denoted by capital letters such as F or L2 . Particularly, if we want to emphasize the concrete nature of a functor we use notation F : (C, U ) → (D, V ) utilizing concrete notations for categories. In F I particular an upright bold letter in case of a functor such as F : C − →D→ − Cat, specially denotes that it is a composite functor (post composing general F with trivial categorifying functor I) involving new concept of trivial categorification. We use blackboard bold (double-struck) letters for composite functor (post composing general F F with underlying functor or concrete functor U : (D, U ) → (Set, ID)) F : C − → U D− → Set where U is the underlying faithful functor of the concrete category (D, U ) over Set. We use ‘pseudo-functor on B’ to refer to a contravariant pseudo-functor ¯ : Bop → Cat. Ψ : B → Cat. This is thought of as a covariant pseudo-functor Ψ It is often notation-wise abused and denoted simply as Ψ : Bop → Cat. Then the Grothendieck Rconstruction (original version) of this contravariant pseudo-functor Ψ, is denoted as B Ψ. (F, C, D) (F, R C, Set) ¯ F Cop X oF C X R oF C ¯ F Cop D→ f I Meas Measure LocMeas LocMeasure
Abstract graph of a functor Concrete graph of a functor Abstract right action of C on objects in F C as trivial categories Abstract left action of C on objects in F C as trivial categories Concrete left action of C on underlying sets of F C Concrete right action of C on underlying sets of F C Arrow category of category D Restriction (f |I ) of function f to domain I Category of measurable spaces (X, ΣX ) and measurable maps (functions) f : X → Y Category of measure spaces (X, ΣX , µ) and inverse-measure-preserving morphisms f : X → Y Category of localizable measurable spaces (X, ΣX , N (µ)) and f • a.e. equivalence classes of non-singular measurable maps f : X → Y Category of localizable measure spaces (X, ΣX , µ) and f • a.e. equivalence classes of inverse-measure-preserving maps f : X → Y xvi
NOTATION AND TERMINOLOGY
xvii
compBoolAlg Category of Dedekind σ-complete Boolean algebras B and sequentially order-continuous Boolean homomorphisms π : B → A MeasureAlg Category of measure algebras (B, µ ¯) and SOC measure-preserving Boolean homomorphisms π : B → A countMeas Subcategory of Meas (or LocMeas) of measurable spaces (X, PX) (or (X, PX, φ))and measurable maps (functions) f : X → Y countMeasure Subcategory of Measure (or LocMeasure) of measure spaces (X, PX, count) and inverse-measure-preserving maps f : X → Y Riesz Category of Riesz spaces and Riesz homomorphisms BanLatt Category of Banach lattices and bounded Riesz homomorphisms Hilb Category of Hilbert spaces and continuous(bounded) linear maps compBoolAlg Category of Dedekind σ-complete Boolean algebras B and sequentially order-continuous Boolean homomorphisms π : B → A 0 L (X, ΣX ) (or L0 = L0 (µ)) Space of virtually measurable real-valued functions f defined on conegligible subsets of X 0 Space of measurable real-valued functions f defined on X LX p L (X, ΣX ) (or Lp = Lp (µ), p ∈ (1, ∞)) Set of functions f ∈ L0 = L0 (µ) such that |f |p is integrable L0 (X, ΣX , µ) (or L0 = L0 (µ) = L0 (X, ΣX , N (µ))) Set of equivalence classes f • in L0 (µ) under =a.e. Lp (X, ΣX , µ) (or Lp = Lp (µ), p ∈ (1, ∞)) Set of functions {f • : f ∈ Lp } ⊆ L0 = L0 (µ) in L0 (µ) under =a.e.
c 2018, Indian Institute of Technology Delhi
Chapter 1
Introduction
Signal representation lies at the heart of how information is represented in signals making it fundamental to many broad range of applications [Kay93] such as high fidelity music reproduction, communications, medical imaging, speech processing, radar and sonar, and oil prospecting. This also makes it intertwined with modern representation learning [BCV13] which deals with the problem of effective data representation. Classically a signal is viewed as an entity varying naturally in time or space and modeled as an element of a linear function space. Fixed mathematical structures on complete domain (Rn ) such as measure and topology along with symmetry are exploited by invoking a suitable group action on the signal space; thereby utilizing the techniques from functional and harmonic analysis developed in the last century; refer [HW89] and references therein. The generative theory of Leyton [Ley01] argues, that in human psychology various shapes and objects such as sheet music symbols or their aggregates forming melodies are highly structured with maximal transfer of previously occurring objects. The theory also intuitively demonstrates that in general various naturally generated shapes all have these kind of structures. Motivated by this generative intuition in this work we propose a philosophy that a functor can be thought of as a (generative) system which connects abstract cause structured as an input category with observed concrete effect also structured as an output category as shown in Figure 1.1, Cause
Functor
Effect
Figure 1.1: Functor as a system connecting abstract cause with concrete effect.
1.1 Background and motivation
2
In context of signal representation one can utilize the mathematical structure of a groupoid1 as an input to model abstract objects carrying certain structures or properties and isomorphisms as their transfer. A detailed heuristic discussion demonstrating advantages of groupoid and fibration over wreath products in context of generative theory [Ley01] is done in Chapter 5 of thesis. In Section 1.1 we summarize the motivations for this work while in Section 1.2 we give an overall structure of the thesis.
1.1
Background and motivation
In this section, we discuss some background and motivations for the proposing functorial signal representation along with the closely associated family of base structured categories characterizing a functor in multiple ways.
Music sheet to concrete waveform Piano
Sensor
Figure 1.2: Functorial system transforming abstract structured melodies to concrete music signals. Referring Figures 1.2, 1.3 and 1.4 we think of the shaded block comprising of generating and sensing system as a functor. It transforms a groupoid (or more generally a category) underlying a generative structure into a concrete signal as image subcategory. More precisely it is modeled as a faithful isomorphism-preserving functor F : C → D. As an example if we denote sheet music symbols (or their aggregates 1
An abstract category C consists of a collection X, Y, Z... of objects denoted by Ob(C), for every pair X, Y ∈ Ob(C), a collection C(X, Y ) = {f : X → Y | X, Y ∈ Ob(C)} of morphisms, for each X ∈ Ob(C), an identity morphism idX or 1X : X → X and composition C(Y, Z) × C(X, Y ) 7→ C(X, Z), i.e (g, f ) 7→ g ◦ f , satisfying the unit law for a morphism f : X → Y , idY ◦ f = f = f ◦ idX f
g
h
and usual associativity for X − →Y − →Z− → W , h ◦ (g ◦ f ) = (h ◦ g) ◦ f . A groupoid is a simply a category in which every morphism is invertible.
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1.1 Background and motivation
3
Reflected light to concrete waveform Reflected light from object
Physical object
Sensor
Figure 1.3: Functorial system transforming reflected light with object structure to concrete image signals.
Vocal utterance to concrete signal Speech formulation
Linguistic word
Sensor
Figure 1.4: Functorial system transforming words to concrete speech signals. forming melodies) as objects G1 , G2 , G3 , ... with transfers as a1 , a2 , ... forming category C then signal generation mechanism becomes functorial as shown in Equation 1.1 where the individual output waveforms are represented by F G1 , F G2 , F G3 , ... while their preserved relationships are captured by arrows F a1 , F a2 , ... forming category F C. *
G1
idG1
a3 ·a2 ·a1 idG4
*
G4 o
a1
a2 ·a1 a3
/
G2 &
t
idG2
a2
G3
t
idG3
4
F a1
idF G1F G1 F a3 ◦F a2 ◦F a1
4
idF G4F G4
o
F a2 ◦F a1 F a3
j
/
F G2idF G2
(1.1)
F a2
'
j
F G3idF G3
Practically the natural generators are very complex structured abstract objects, thought of sitting in multiple categories of various structures. Depending on a particular structure of interest within the observed signal, a concrete codomain category is chosen. Then the functor F is constrained to be faithful and isomorphism-preserving.
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1.1 Background and motivation
4
Being faithful and isomorphism-preserving ensures that using isomorphisms in the category F C we can uniquely infer transfers (isomorphisms) between generators (objects) thereby directly gaining insight into generative mechanism of source. In this work, we shall be particularly interested in measurable and measure-preserving structures motivated by translational, scaling, amplitude redundancies very common in classic image, audio and speech signals. This interest is also attributed to connections with classical representation techniques in spaces of measurable and square integrable functions such as L2 (Rn ). Often in real world scenarios the equipments are never ideal and further the superposition of waveforms in observed signal makes the functor non-faithful. This puts certain limitations on groupoid in the domain category that could be inferred from the observed signal which we study in Chapter 4 along with solutions and work-around. Interestingly by using the classic set-theoretic measure theory in addition to pure category theory certain limitations could be effectively overcome by using novel concept of trivial categorification; proposed in Chapter 3. Loosely speaking this treats objects as trivial categories and therefore we can utilize additional set-theoretic properties of objects independently along with treating them simply as objects of enclosing category. As an example, in the context of this work a measurable function f : (I, ΣI ) → (R, ΣB ) which is an object of Meas→ is also a trivial category and therefore by considering additional property of R being a field, it is can be point-wise added to and multiplied by any other measurable function g : (I, ΣI ) → (R, ΣB ). In other words, f is simultaneously an element (settheoretically) of Riesz space L0 I and this property is independent of it being an object (category-theoretically) of Meas→ which recognizes only measurable structure on R. This novel concept of using set-theory alongside category-theory simultaneously for application is explored in this thesis. The proposed mathematical model of a functor for natural generating mechanism of a signal offers various additional benefits when compared to classic representation techniques: • Signal source as base category: In classic case, the mathematical structure of a source generating a signal carrying information of interest is usually not taken into consideration. From the perspective of information theory it is treated as either memoryless or with memory. By modeling a source with memory as a groupoid in tune with generative intuition we seek to capture isomorphic c 2018, Indian Institute of Technology Delhi
1.1 Background and motivation
5
relationships between waveforms generated by the source directly impacting the amount of perceived information in signal. The memoryless source as a special case having no interdependencies of successive messages is modeled as a discrete category. • Redundancy using relative perspective: Classic redundancy is defined as ratio of actual rate of source to its absolute rate and indicates the maximum possible data compression ratio by which bits in its efficient representation can be decreased. In functorial model, the relative perspective offered by category theory provides an authentic tool to model interdependence between messages or sub-signals. This leads to arrow-theoretic structural definition of redundancy. It also becomes possible to understand compression in a natural category-theoretic way. • Signal space as a category: The generic signal or message spaces such as L2 (Rn ) cater mainly to memoryless sources since the linearly independent basis best model sub-signals or messages which are independent. In functorial model observed signal becomes an image subcategory inducing additional category structure on domain such as Rn . The Signal space in certain cases can be modeled as a category matched to the generative structure of the signal and therefore unique to every signal. Next we summarize two natural motivations for introducing the concept of base structured categories. 1. Functorial signal representation and precise arrow-theoretic mathematical formulation of redundancy within signals: Fibred Category Pure category theory Arrows decomposed into vertical and Cartesian Arrows treated fundamental Complete categorification (D treated as 2-category) Fibres are pure categories
Base Structured Category Partly category, set-theory (objects with properties on underlying sets) All arrows Cartesian (trivial vertical arrows) Objects treated in set theory while arrows depict relativity Partial categorification (D trivially categorified) Fibres are objects viewed as trivial categories
Table 1.1: Heuristic of base structured category combining category theory with set theory
A surprisingly trivial construction (F, C, D) (being very obviously isomorphic to category C) for the purposes of this work offers a distinctly refreshingly new c 2018, Indian Institute of Technology Delhi
1.1 Background and motivation
6
perspective on any ordinary functor F : C → D. It is the mathematical expression which combines the intuitive generative perspective of Leyton’s Theory in psychology [Ley86a], [Ley01] along with well-known Grothendieck’s relative point of view [nLWd] by treating it as a fibration. It leads to formulation of category (F, C, Set) which can be viewed as opfibration. The distinct perspective in a nutshell is that while general fibred categories carry both structures i.e from base category as well as fibre categories through Cartesian and vertical arrows, the base structured categories carry only the base structure concealing the vertical structure within the objects since vertical arrows are just identities. By considering additional properties on the underlying sets of objects in D which are transparent to usual category structure on objects in D we can treat objects in F C also set-theoretically. It is our conjecture that this can be particularly useful in situations where the category D is notoriously difficult to work with (such as having bad properties) but unavoidable in applications. This perspective is utilized in the context of signal representation where D is Meas→ . By considering properties of field on codomain measurable space R the measurable functions are also viewed as elements of linear spaces. We can also view a signal as being fibred on some category C that captures natural generative mechanism of a given signal to be represented. Then intra-signal redundancy gets naturally modeled as Cartesian lift of the base isomorphisms. Going by the philosophy of category theory which treats arrows as more fundamental to objects these categories exploit the generative base C structure of a signal explicitly in category theory while the vertical Meas→ structure (often Riesz space by additional properties on underlying sets of Meas→ objects as described) hidden from categorical formulation can be treated in set-theory independently at the local level of objects utilizing benefits of both for applications. In essence the signal can be treated as a category (F, C, D) emphasizing the generative structure of a signal. 2. Transformation Categories generalizing transformation groupoid X//G In the spirit of a group action on a set giving rise to a transformation groupoid; the base category action on the underlying sets of RD-objects (in case of D ¯ which capture the being construct) gives rise to categories X oF C and Cop F concrete action of base arrows on the underlying elements of the D-objects. These base structured categories are then motivated as the generalization of classic transformation groupoids suggesting the terminology Transformation Categories. Now the semidirect product of groups (which is a group), semidirect product of monoids (which is again a monoid), semidirect product of categories (which c 2018, Indian Institute of Technology Delhi
1.1 Background and motivation
7
is again a category) and of a set (treated as a discrete category) and a category (which obviously results into a category often termed as transformation groupoid). All of these are just semidirect products of purely categories since the structures of a group, monoid, set are all modeled as a category. But what about the case of objects of an arbitrary category D which cannot be easily treated as categories? It is visually seen using commutative diagrams that even in an arbitrary case of a functor F : C → D clearly there is an induced action of arrows. This is the motivationRfor introducing new meaningful mathematical ¯ which make implicit action in a functor expressions such as X oF C and Cop F precisely explicit at least partly by treating objects as trivial categories. The action at level of object is only partially captured by category theory in the sense that object after being transformed by arrows is still viewed as same object using trivial category. However note that when the object undergoes an internal state transformation this change can be captured at level of elements within the object considered as structured set. Thus realizing that actions at global (or outer level) are being captured category-theoretically while actions at local (or inner) object level implicitly remain at set-theoretic level one actually makes use both these theories simultaneously for application purposes especially in symmetries and geometries. This motivates us to interpret a general functor F : C → D as an action; Visually we have I / F / D Cat C f
X g◦f
/
Y
FX g
F
/
Ff
/ #
FY
F g◦F f
Z
FX
Fg
I
/
Fg◦Ff
FZ
Ordinary functor as action of category on objects F : C → D as F : C → D → Cat
Ff
/ "
FY
Fg
FZ
Category-theoretic Semidirect Product (FX q FY q ...) oF C
Table 1.2: Category C action on objects F X, F Y, ... within image subcategory FC
While the action on underlying sets of objects within image subcategory FC is motivated by functor F : C → D → Set as an action; Visually we have C
F
/
D
U
/
Set
c 2018, Indian Institute of Technology Delhi
1.2 Structure and contributions
f
X g◦f
/Y
8
FX g
Z
F
/
Ff
F g◦F f
/ #
FY
FX U
Fg
/ Fg◦Ff
FZ
Functor as action of category on underlying sets of objects F : C → D as F : C → D → Set
Ff
/ "
FY
Fg
FZ Category-theoretic Semidirect Product (FX q FY q ...) oF C
Table 1.3: Category C action on underlying sets of F X, F Y, ... within image subcategory FC
In essence this perspective motivates us to model a Klein geometry partly category theoretically as X oF H where manifolds are trivially categorified bringing in a functor underlying geometries. Such a mathematical formulation which works for any arbitrary category D (in this context category of manifolds) could lead to possibly new formulation of groupoid geometries combining category theory and set-theoretic manifold theory as some sort of multi-object generalization of Klein group geometries since the major hurdle of existence of coproduct objects in D is eliminated by trivially categorizing objects. This is described in Chapter 5 and considered as spin-off of the main work. In summary, the transformation categories denote the action of C category arrows on arbitrary D-category objects lying within image subcategory F (C); thus every action in mathematics becomes mathematically expressible in form of categorical semidirect product.
1.2
Structure and contributions
In this section we present the overall structure of thesis and summarize some important results and contributions. The outline of the thesis is as follows: • Chapter 2 basically is on the prerequisites from basic category theory mainly referring [BW90] [JAS], fibred category theory referring [Jac99] [Gro71], measure theory and functional analysis referring [Fre11], [Fre16], [Fre12], [Fre13]. Section 2.1.1 recalls the basic definitions of category and functor additionally with their concrete counterparts. Section 2.1.2 is on product and pullback as c 2018, Indian Institute of Technology Delhi
1.2 Structure and contributions
9
special cases of a general concept of limit. Section 2.2 recalls concepts from fibred category theory such as set-indexed sets, base change, fibration along with some of its simple properties, cloven and split fibrations, indexed categories along with fundamental construction called Grothendieck construction. Then we visually demonstrate connection of category (F, C, Cat) to fibred category. Section 2.3.1 recalls the basic definitions of σ-algebra, measurable and measure space along with a result dealing with the partial order structure on equivalence classes in σ-algebra. Next Section 2.3.2 discusses the concepts of Boolean and measure algebras thought of as category theoretic duals to measurable and measure spaces. Further in Section 2.3.3 we recall the concepts of point-supported and counting measures using atomic spaces and algebras. Next Section 2.3.4 is on various types of measurable functions and measurepreserving functions along with Boolean homomorphisms. Finally Section 2.3.5 recalls certain additional results required for this work such as composite measurable function, basic properties of measurable functions, coneglibility and virtually measurable functions. • Chapter 3 explains the novel concept of “base structured categories” characterizing a functor in various ways. Section 3.1.1 formulates and discusses definitions characterizing a functor as structure preserving morphism such as (F, C, D) as an abstract graph of a functor and (F, C, Set) as a concrete graph of a functor. Here we also prove new Propositions 60, 62 on these categories taking the perspective of fibration on them. Section 3.2 formulates a precise concept of Trivial categorification of objects in D category required to make sense of the fibration perspective in previous section as special case of the concept of categorification [JCB98] and formulate new definitions characterizing a functor as an action R of category on the ¯ as an abstract family of objects within the imageR subcategory such as Cop F ¯ as concrete right action induced by a right action induced by a functor, Cop F functor, X oF C as abstract left action induced by a functor, X oF C as concrete left action induced by a functor. Then we describe the duality between categories defined as right and left actions. Next we prove a Proposition 71 using transformation groupoid [JCM14] justifying the terminology of transformation categories for them. Finally we prove Proposition 72 describing the interrelationship between base structured categories. • Chapter 4 explains the novel proposition of “Functorial Signal Representation” along with signal redundancy in an arrow-theoretic manner. Additionally by c 2018, Indian Institute of Technology Delhi
1.2 Structure and contributions
10
collecting various forms of L0 and L2 functors scattered across vast literature of both measure theory and category theory we utilize these in studying elementary properties of signal spaces as categories. Section 4.1 utilizes the perspective of functor as a structure preserving morphism in the context of signal representation to propose a new functorial model for signals as measurable functions and introduce the concept of Signal (matched) Space as a category. Table 4.1 summarizes the differences between contemporary and new functorial model, simultaneously showing new framework as generalization of the classic model which is discussed in detail in subsequent subsections. Subsection 4.1.9 discusses alternate 2-category perspective of signal as subcategory of functor category Meas2 . Further in Subsection 4.1.10, we discuss general limitations including practical observation errors and partial isomorphisms between the generators along with some examples. Section 4.2 contributes a novel structural definition 73 of redundancy and some of its special cases of translational redundancy affine redundancy and affine amplified/attenuated redundancy using the concept of isomorphism in category theory. Using it we illustrate how compression occurs in a differential encoding schemes by taking a real world compression standard of PNG and Lossless JPEG by casting the de-correlating DPCM and predictive coding stage of all lossless image compression standards as a special case of the proposed framework. In Section 4.3 we study five relevant categories and their duals with objects and arrows as summarized in Table 4.12 and Table 4.13 used in formulating various forms of L0 and L2 functors.The study leads to two Theorems 78, 79 of particular importance already in measure theory, which are useful to establish duality between these categories. They are not new contribution except for consolidating and defining constructions L0 , L2 , l0 , l2 as functors concisely all possible forms in the context of signal matched spaces as categories. In Section we briefly formulate different codomain categories depending upon the kind of structure one is interested on signal space as summarized in Table 4.3.4. In Section 4.4 using Propositions 81 and 86 we explicitly defining these functors directly using measurable or measure spaces without invoking the dual concept of Boolean and measure algebras.Using these proposition we explicitly form various inter-related definitions of L0 ,L2 ,l0 ,l2 both covariant and contravariant. Not all of these definitions are new (except the case L0 Functor extended to partial category Par(LocMeas, M) but useful for the functorial framework we have proposed. The (contravariant) functor L0 : LocMeas → Riesz, the (contravariant) functor l0 : countLocMeas → Riesz, the (covariant) functor L0 : compBoolAlg → Riesz. The (contravariant) functor c 2018, Indian Institute of Technology Delhi
1.2 Structure and contributions
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L2 : LocMeasure → BanLatt, the (contravariant) functor L2 : LocMeasure → Hilb, the (contravariant) functor l2 : countLocMeasure → Hilb, the (covariant) functor L2 : MeasureAlg → BanLatt. The (contravariant) functor L0 : Par(LocMeas, M) → Riesz generalizing (covariant) functor l2 : PInj → Hilb. Finally we summarize all functorial models and their dual versions concisely in Table 4.15. Finally Section 4.5 discusses some standard properties such as linearity, lattice ˆ B , N (µ)), L0 along with adand multiplication of classical signal spaces L0 (R, Σ R ˆ B , µ). In general all these properties ditional norm and inner-product of L2 (R, Σ in subcategory of Riesz or BanLatt are simply local or valid on objects within this category. The structure-preserving morphisms in these subcategories are linear operators preserving these properties considered as structures on objects. • Chapter 5 deals with concept of transformation categories as applicable to symmetry and geometry. This is considered as spin off of work in earlier Chapter 3. This chapter more or less heuristically discusses the principal reason for using groupoid instead of wreath group for dealing with hierarchy of symmetries. The main references for this work (still in progress as it also requires expertise from manifold theory) are [Car09], [Wei96], [Bro87]. Section 5.1 deals with category action and sets viewed as categories while Section 5.2 is on global symmetry of a homogeneous structure using a single object base groupoid along with local and global symmetries of a non-homogeneous structure using a multi-object base groupoid. Section 5.3 deals with modeling Leyton’s generative theory using groupoids. Section 5.4 compares hierarchy of symmetries using hierarchy of Groups [(X oF2 (G3 × G3 )] oF1 G2 against hierarchy of Groupoids [(X2a oF2a G3 ) q (X2b oF2b G3 )] oF1 G2 and connection of hierarchy of base structured categories with standard composite of Grothendieck fibrations. Next Section 5.5 discusses the naturality square corresponding to inner 2-group action along with the naturality square at a node of outer naturality square and then naturality square corresponding to outer 2G action. Further in Section 5.6 namely Geometries as Base structured categories X oF C proposes viewing the geometries using transformation categories. Finally Section 5.7 is on heuristic differences between single object and multi-object species of structures. • Chapter 6 finally briefly summarizes the philosophical implications of this work in a broader context and various directions for applying and improving upon the proposed framework towards unifying of various representation techniques along with a systems approach.
c 2018, Indian Institute of Technology Delhi
1.2 Structure and contributions
The work in thesis is partly published as [Sam16], through, [SJb], [SJc] and [SJd].
12
[SJa] and under review
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Chapter 2
Preliminaries
In this chapter we introduce all required preliminaries that we shall use in the rest of this thesis. There are three major sections covering basic and fibred category theory along with elementary measure theory and some functional analysis. Section 2.1 is devoted entirely to fundamental category theory while the required concepts from fibred category theory are recalled in Section 2.2. Section 2.3 caters to measure theory, measure algebras from a categorical viewpoint which we have used in Chapter 4. It also covers some elementary concepts on functional analysis.
2.1 2.1.1
Category theory and Grothendieck fibration Category and functor
Having introduced by Eilenberg and Mac-Lane first in [EM45]; the standard definition of a category and a functor has evolved somewhat to a form mostly found in today’s standard reference text such as [Mac98]. For an excellent historical account of this field with an intuitive approach, the reader is referred to [Mar09]. Roughly speaking structures of a particular type with morphisms preserving this structure form a category.
Definition 1 [Mac98] An abstract category C consists of: • objects: a collection X, Y, Z... denoted by Ob(C)
2.1 Category theory and Grothendieck fibration
14
• morphisms: for every pair X, Y ∈ Ob(C), a collection C(X, Y ) = {f : X → Y | X, Y ∈ Ob(C)} • identity: for each X ∈ Ob(C), a morphism idX or 1X : X → X • composition: C(Y, Z) × C(X, Y ) 7→ C(X, Z), i.e (g, f ) 7→ g ◦ f • unit laws: for a morphism f : X → Y we have idY ◦ f = f = f ◦ idX f
g
h
• associativity: for X → − Y → − Z→ − W we have h ◦ (g ◦ f ) = (h ◦ g) ◦ f .
The schematic representation of an abstract category will be shown as idX
(
X
h◦g◦f idW
)
Wo
f
/
w
Y
idY
g g◦f h
& w
Z
idZ
Just like structures form categories of various sorts, categories themselves are also structures of a particular type with functors as morphisms. Indeed they form a category denoted as Cat. A functor makes coherence between different structures precise and this coherence of transformation is widely referred to as functoriality.
Definition 2 [Mac98] Suppose C and D are categories. A map F : C → D is a (covariant) functor consisting of: • object map: to every X ∈ C, an object F (X) ∈ Ob(D) • morphism map: to every morphism f ∈ C(X, Y ), a morphism F (f ) : F (X) → F (Y ) • identity map: for all X ∈ C, F (idX ) = idF (X) f
g
• composition map: for all morphisms X → − Y → − Z F (f ◦ g) = F (f ) ◦ F (g).
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2.1 Category theory and Grothendieck fibration
15
The schematic representation of a functor will be shown as idX
(
X
/
w
Y
0
idY
g
h◦g◦f idW
f
)
Wo
g◦f h
& w
Z
Ff
idF XF X F h◦F g◦F f
2
idZ
idF WF W
o
F g◦F f Fh
p
/
F Y idF Y Fg
'
p
F Z idF Z
Remark 3 The concise schematic diagrams included along with the definitions are meant to graphically illustrate the various axioms stated. These are quite commonly known by the term ‘commutative diagrams’. Going forward in the thesis, we will make extensive use of these such diagrams. Otherwise and often difficult to grasp deep abstract concepts are simply made visually concrete and intuitive with their usage. Definition 4 [JAS] Let X be a category, then a concrete category over X is a pair (C, U ), where C is a category and U : C → X is a faithful functor. Often U will be called the underlying functor of the concrete category and X the underlying category for (C, U ). In the standard category literature [JAS] the underlying category is commonly called the base category which we avoid for obvious reasons. Concrete categories over Set are called constructs which are precisely the categories of structured sets and structurepreserving functions between them. Definition 5 [JAS] Let (C, U ) and (D, V ) be some concrete categories over X, then a concrete functor from (C, U ) to (D, V ) is a functor F : C → D with U = V ◦ F and denoted as F : (C, U ) → (D, V ). The condition U = V ◦ F implies that U (X) = V (F X) (where X is an object of C) and U (f ) = V (F f ) (where X is an object and f is morphism of C). Thus in the case of constructs, concrete functors ensure that the underlying sets of objects and set functions of morphisms in C are in bijection with F C. This intuitively means that the information concerning the underlying sets and functions of objects and morphisms is also preserved by the concrete functors. This is distinctly in contrast c 2018, Indian Institute of Technology Delhi
2.1 Category theory and Grothendieck fibration
16
to an usual abstract functor F : C → D between the concrete categories (C, U ) and (D, V ) which need not preserve the structure of underlying category X. Definition 6 [JAS] A functor F : C → D between categories is called an isomorphism if there is a functor G : D → C such that G ◦ F = idC and F ◦ G = idD .
2.1.2
Product, pullback and limits
Before we can assimilate the theory of fibration, we need to revisit the fundamental notion of limit in category theory.Limits, and colimits, the dual concepts are intimately connected to the ideas of universal property and universal contructions. Since the definition of limit is quite general, it is easier to formulate it by first studying some of its special cases: products and pullbacks. These examples of limits occur throughout thesis and are required especially in the formulation of basic fibred categories. We begin with a brief recall of product which is one of the simplest limit to grasp. Then we move on to define pullbacks as limits which are extremely fundamental to our entire work. Pullbacks are also directly related to the central concept of cartesian lifts utilized in the theory of fibration. Finally we discuss the actual concept of limit in an intuitive fashion using motivation from [Mil14] in addition to precise unified definition using the approach from [Lei14]. The standard reference material for these limits is [Mac98],[Lei14],[LS09].
Product The simple cartesian product of two arbitrary sets is well known. This construction is a limit in disguise and provides an example of categorical product in the category Set. The reader especially from signals and information theory background is strongly encouraged to refer [LS09] to get familiarized with basic constructions such as the cartesian product and overall set theory viewed from the perspective of category theory using the conceptual arrow approach. Here we simply recall the standard definition of product in the category theory to motivate the immediately following general concept of limit. c 2018, Indian Institute of Technology Delhi
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Definition 7 ([Mac98]) Let C be a category and let A, B, C, N ∈ C then a product of A and B consists of an object C and projection maps p1 , p2 with the property that for all objects such as N with maps q1 , q2 in C, there exists a unique map h : N → C such that p1 ◦ h = q1 and p2 ◦ h = q2 . The maps p1 and p2 are called the projections and we write C as A × B. The uniqueness of h is often referred to as the universal property of a product.
Example 8 Consider two categories A, B which are objects of a category Cat. This is a category with objects as small categories and arrows as functors. The category A × B is a category with objects the < a, b > pairs of objects a of A and b of B. The arrow < a, b >→< a0 , b0 > is the pair < f, g > of arrows f : a → a0 , g : b → b0 from A and B respectively. It can be easily verified that this indeed is a category where everything is done component-wise Further it satisfies the universal property of a product as defined earlier.
N
q2 h q1
#
A×B
p2
/% B
p1
A
Figure 2.1: Universal property of a product
Pullback It is extremely crucial to get a sound understanding of the concept of pullback (a special case of limit where J = (• → • ← •) is a small category with three objects and two arrows other than identities) both heuristically as well as rigorously since the delicate concept of cartesian lift is completely based on this limit concept. Hence we shall also consider simple additional examples following the definition.
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Definition 9 [Mac98] Let J be a small category with three objects and two arrows other than identities represented as J = (• → • ← •), then a diagram D : J → C f
g
/Co is a pair of arrows A B with a common co-domain C. The cone over this diagram is a pair of arrows from a vertex N such that the resulting square shown below (left) commutes. Then a universal cone or a limit with vertex P = A ×C B satisfies the universal property with a unique h : N → A ×C B as shown. The square formed by this universal cone is called a pullback square and its vertex is called a pullback, a ‘fibered product‘, or a product over (the base object) C. Alternately it is said that f arises by pulling back along g,and g 0 arises by pulling back g along f .
N
(2.1)
q2 h q1
$
A ×C B
p2
=f 0
&/
p1 =g 0
B
A
g f
/
C
The inner square formed in (2.1) is called a pullback or cartesian square. Here if we compare the commutative diagram with that of a product then there is an additional vertex which has a non-trivial object C called base which gives it a significant power. Indeed if C is a terminal object in the particular category in which commutative square (2.1) is considered then pullback precisely is the common product where we might drop the vertex altogether since the maps f, g are unique without altering the essential definition. Let us absorb the heuristic that a pullback object P holds a relationship with object B precisely in a way the object A is related to the object C by considering examples now.
Example 10 In the category FinSet the objects are finite sets and arrows are the usual set functions. Then the pullback is P = {(a, b) : a ∈ A, b ∈ B, f (a) = g(b)}. Thus P is a particular subset of the cartesian product set A × B. Intuitively as shown in the Figure 2.2 this can be interpreted as C indexing and partitioning both A and c 2018, Indian Institute of Technology Delhi
2.1 Category theory and Grothendieck fibration
P
19
A
p2
a1 a2
p1
b1
f
c1 b2 b3
c2
g
B
C
Figure 2.2: Heuristic of a Pullback in FinSet B and then P is formed by the cartesian product of elements of A and B partitionwise or respecting the indexing structure. Hence note that the elements of P are in the same proportion relative to elements in B, which is precisely the proportion of elements of A relative to elements in C. Thus the heuristic that the pullback object holds a relationship with object B precisely in a way the object A is related to the object C takes concrete form of relative count of elements in this example.
Example 11 In the special case following Example 10, where the g is an inclusion function, p1 or f 0 can be viewed as an appropriate restriction of f as shown in Figure 2.3. Here P = f −1 (B) = {a : a ∈ A, f (a) ∈ B}. This is the usual inverse image of a function f −1 (B) _
f0
A
f
/
/
B _
C
Figure 2.3: Inverse image of a function as a pullback
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Limits Having looked at the constructions: product and pullbacks, the common theme tying them is the concept of a limit. An overall intuitive, geometric or physical interpretation of a limit is that it embodies the structure or properties of a given diagram D completely in a single object which contains exactly the same amount of information about the whole diagram neither more nor less. However the precise technical definition of this concept in classic reference [Mac98] utilizes the concept of natural transformation. Hence one might wish considering [Mil14] for more visual treatment of this concept. However if the reader is already familiar with concept of natural transformation then the following description recalls limit in a classical manner. In each construction starting with some objects some maps between them, we seek a new object together with maps from it to the original objects, with a universal property resulting into a situation often called a universal construction. The new object is universal amongst the objects that have a certain property.
Definition 12 [Lei14] Let C be a category and J a small category. A functor D : J → C is called a diagram of shape J in C. 12 N fX
/
w
X u
&
v
Y
?Z
+
DX
fZ fY
}
Dw
Du
'+
/+ DZ ; Dv
DY
Figure 2.4: A functor D and a cone with vertex N In general we could consider a category J which is small and often finite to be an abstraction of some finite pattern or structure or shape. Then a functor D from J to some another category C is termed as J-shaped diagram in C. It is intuitively thought of indexing some collection of objects and morphisms in C patterned on J. c 2018, Indian Institute of Technology Delhi
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A constant functor ∆N from J to C sends every object to N and every morphism to idN . The natural transformation between these from ∆N to D is called a cone with vertex N . This is shown in Figure 2.4 and visually resembles a cone, since the image of ∆N is the apex of a pyramid or cone whose sides are given by the components of the natural transformation with the image of D forming the base of that cone.
Definition 13 [Mac98] A limit of a diagram D is the universal cone (or limiting cone) of a diagram D. N
∃f¯!
LimD
fX pX
fZ pZ
fY pY
|
/
Dw
DX
Du
(
DZ :
Dv
DY
Figure 2.5: Limit of a diagram D
The Definitions 12, 13 could be unified in to give a single comprehensive definition of limit following [Lei14] without referring to the concept of natural transformation explicitly.
Definition 14 [Lei14] Let C be a category, J a small category, and D : J → C a diagram in C. Then, 1. A cone on D is an object N ∈ C (the vertex of the cone) together with a family fI A −→ D(I) (2.2) X∈J
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u
of maps in C such that for all maps X −→ Y in J, the triangle fX
N fY
DX 6 (
Du
DY
commutes. 2. A limit of D is a cone
p
X LimD −→ DX
with the property that for any cone (2.2) on D, there exists a unique map f¯ : N → LimD such that pX ◦f¯ = fX for all X ∈ J. The maps pX are called the projections of the limit. X∈J
It is easy to see that the product discussed earlier in Section 2.1.2 is a simple example of limit with J = (••) is a small category with two objects and no arrows except two identities.
2.2
Fibred category theory
Having recalled the basic category, functor and pullback we are ready to study the fibred category theory for our work. Although the original reference is [Section VI [Gro71]] for us the easier references have been [Jac99], [Vis04]. We also have gained much from [BL09]. This section deals explicitly with the traditional fibred category theory. The formal theory arise from the work of Grothendieck in algebraic geometry and deals with indexing of category using another category. In the theory there are two equivalent formulations viz. category-indexed category and categorical fibration. They are a natural generalization of the concept of set-indexed set to categories, hence it will be easier for us to first consider briefly the essence of set-indexed set for our work.
2.2.1
Set-Indexed Sets
There are in general two different equivalent ways of presenting the concept of setindexed sets as studied in [Jac99]. c 2018, Indian Institute of Technology Delhi
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1. Pointwise or Split Indexing: In this form of indexing, to every element i ∈ I some ordinary set Xi is assigned. In the language of category theory, this can be formulated using a functor Ψ : I → Set. Notice that since in the category Set (of all small sets with functions) every singleton is an object, we can form a functor from a (discrete) subcategory I with objects corresponding to the elements of set I to the category Set. Note that by the definition of functor the image (category) will precisely contain objects Xi . 2. Display Indexing: This is simply given by an ordinary function f : X → I. Heuristically this decomposes or partitions the whole set X into particular subsets, which is specified by f .The subsets are termed as fibers over the elements of I.
It is important to realize here that the specification of f in turn specifies how X is viewed structurally relative to I. Since this particular case deals only with objects, with all arrows trivial, it only specifies how X is partitioned or more precisely X = qi Xi where the union is necessarily disjoint. However with generalization to arbitrary arrows or general categories it has a far reaching powerful interpretation on how structures are viewed relatively. To forget the indexing or base in this case, simply amounts to forgetting the particular partition induced by this index set. Further display indexing is simply characterized by a general set function in which fibers are always disjoint (being inverse images of this function), it readily generalizes to categories as compared to point-wise indexing. The function displays the family over a base, hence the terminology. Example 15 The constant family (or trivial fibration) consists of all fibres being isomorphic to one set Xi . Example 16 The arrow category Set→ which contains set-indexed sets or a family as objects with morphisms preserving the indexing structure. The category Set→ consists of: • objects: a collection of all set-indexed sets of form
X f I
!
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X f I
• morphisms: for every pair
!
!
Y
a morphism (u, h) where u : I → J
g
J
and h : X → Y with g ◦ h = u ◦ f ! • identity: for each
X f I
, a morphism (idI , idX ) or pair of identities from Set
satisfying the usual axioms of a category such as composition, unit laws and associativity. The special case of Set/I widely known as Slice Category is obtained for a fixed index set I.
2.2.2
Change of Base
For a given family,
X f I
! the change of base is achieved using the pull-back operation
in category Set as shown in the Figure 2.6 D
q2 h
q1
Y
J
u0 f0
/
#
X f
u
/I
Figure 2.6: Change of base for set-indexed sets.
Denote this pull-back of f against u as Y which is J ×I X. Given any function f : X → I, we can interpret this as a decomposition of the set X into subsets f −1 (i), indexed by the elements i ∈ I. This way of looking at a function is the essential essence underlying the idea of fibration where X is termed as being fibered on I and f −1 (i) are the fibres of X at i ∈ I. Note that I which will be referred to as base in the general categorical framework, sets up a partition of X as the fibres are necessarily disjoint. c 2018, Indian Institute of Technology Delhi
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2.2.3
25
Fibration
Generalizing the concept of collection of sets varying over an index set to categories, we get the concept of category indexed categories corresponding to point-wise indexing and fibration corresponding to display indexing. The theory of fibred categories makes the concept of collections of categories that vary over a base category mathematically precise. It is also seen as as ordinary category theory over some base category (thought of as a universe). First we consider fibration and later we study category-indexed categories. The fundamental construction called Grothendieck completion that turns category-indexed-categories into fibration is explored thereafter. We begin with recalling the definition of fibration and opfibration. Definition 17 ([Jac99]) Let P : E → B be a usual functor; • A morphism f : X → Y in E is cartesian over u : I → J in B if P f = u and every g : Z → Y in E for which one has P g = uw for some w : P Z → I, there exists a unique morphism h : Z → X in E above w such that f h = g. Z _
g h
E
!
X _
f
*/
Y_
P
PZ
B
u◦w=P g w
!
I
u
*/
J
• The functor P : E → B is a fibration (or a fibred category) if for every Y ∈ E and u : I → P Y in B, there is a cartesian morphism f : X → Y in E above u.
Definition 18 ([Jac99]) Let P : E → B be a usual functor; • A morphism f : X → Y in E is opCartesian over u : I → J in B if P f = u and every g : X → Z in E for which one has P g = wu for some w : J → P Z, uniquely determines an h : Y → Z in E above w with hf = w. c 2018, Indian Institute of Technology Delhi
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4= Z _
g
E
X _
f
h
/
Y_ 4P = Z
P
B
w◦u=P g
I
u
/
w
J
• The functor P : E → B is a opfibration (or a opfibred category) if for every X ∈ E and u : P X → J in B, there is a opCartesian morphism f : X → Y in E above u. This uniqueness resulting in E is termed as the unique lifting property. The concept of cartesian lift is very closely related to pullbacks in ordinary categories. To expose this connection heuristically we use the analogy of mixed categories from [BL09]. For this consider a category in which we mix objects and arrows from E and B as follows: • objects All objects from both E and B i.e X, Y, .. ∈ E and I, J, .. ∈ B • morphisms All f, g, .. ∈ E, u, v, .. ∈ B with PX : X → I where P X = I quotiented by P f = u • identity identities from both E and B • composition usual compositions and cross compositions such as u ◦ PX and PY ◦ f along with usual unit laws and associativity law. Note that by fiat we declare that the (mixed) diagrams of the type shown below truly commute only when P f = u or u ◦ PX = PY ◦ f as shown in Figure 2.7 where PX is the arrow X 7→ I, the usual restriction of the functor P to X whereas f and u are the general arrows in E and B respectively. Note that for f : X → Y to be cartesian precisely means that the resulting square is a pullback or cartesian square in this mixed union category. The usual uniqueness c 2018, Indian Institute of Technology Delhi
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f
X _ PX
I
u
/
/
Y_
PY
J
Figure 2.7: Commutative squares in the mixed category g
Z _ PZ
h
PX
K
/X _
w
/
f
I
'/
Y_
u
PY
/7 J
v
Figure 2.8: Cartesian lift as a pullback in mixed category of this pullback object X could be expressed using the Figure 2.8 which precisely matches the definition of the cartesian lift of u at Y . Hence most of the properties of cartesian arrows which we gather now for our work in the thesis, follow naturally from the usual properties of the pullback in ordinary categories. In particular the intuition for the cartesian lift of a general arrow precisely follows from intuition regarding the pullback we discussed earlier. This essentially says that X relative to Y is exactly similar to I relative to J. In fibration E is thought of as being above B.
2.2.4
Basic Properties of Fibration
Here we recall certain standard results pertaining to the total category E which will be required later in the thesis. Proposition 19 [Jac99] Let P : E → B be a fibration then every morphism in E factors as a vertical map followed by a cartesian one.
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Proof : Consider an arbitrary morphism g : Z → Y in the total category E. Using Definition 2 of a functor P , let P Z = I and P Y = J and P (g) = u where u : I → J. But since P : E → B is also a fibration it follows that Z ∈ EI and Y ∈ EJ and there must exist cartesian lift u at Y . Let this cartesian arrow above u be f then we have a unique h with g = f ◦ h Z g
h f
X
/
Y
But since both f and g lie above u, Z and X must lie above I or in the fiber EI . Thus the resulting unique h is a vertical map. Proposition 19 can be interpreted as saying that the total structure in the fibred category is a fusion of horizontal and vertical structure along with the action of horizontal on vertical. Proposition 20 [Jac99] Let P : E → B be a fibration then composition of cartesian arrows in E is also a cartesian arrow. Proof : Let f and g be the cartesian arrows above some u and v in the base category. Then for some arbitrary arrow j from a fixed object W to Y we have a unique h1 such that f ◦ h1 = j. W h2
Z
~
j
h1 f ◦g g
X
f
*/
Y
But since g is cartesian, for the same fixed object W and arrow h1 to X, we have a unique h2 such that g ◦ h2 = h1 . Noting that all inner triangles commute, we have for an arbitrary arrow j a unique h2 such that (f ◦ g) ◦ h2 = j proving that f ◦ g is cartesian. Proposition 21 [Jac99] Let P : E → B be a fibration then in E Cartesian map c 2018, Indian Institute of Technology Delhi
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above an isomorphism is also an isomorphism.
Proof : Let a B-arrow u : I → J be an isomorphism. This implies by the definition of isomorphism that there exists another B-arrow v : J → I such that u ◦ v = idJ and v ◦ u = idI . Now let the cartesian map above u be f : X → Y for a given Y and the cartesian map above v for the X be g : Z → X where P (Z) = J. Then the vertical map f ◦ g is also cartesian as the composition of cartesian maps is cartesian and since u ◦ v = idJ therefore it must be an identity arrow or f ◦ g = 1Y implying Z =Y. Z g
X
~ f /
f ◦g
Y
similarly in the other direction one can show that g ◦ f = 1X . Thus the cartesian lift of an isomorphism is also an isomorphism.
2.2.5
Cloven and Split Fibration
Since we shall need strict functors rather than pseudo-functors, this section may be skipped without any harm. We include these two kinds of fibrations to get familiarized with the classic fibred category theory associated with pseudo-functors. The definition of fibration guarantees that the cartesian lift (or the pullback completion) for every possible u and Y exists but keeps the choice unspecified. Indeed such a pullback object is only unique up to a unique isomorphism, so one has to explicitly make a choice of cartesian lift. When the choice is made (upto vertical isomorphism) as explained next, the fibration is called cloven and it equivalently corresponds to a pseudo-functor or category-valued presheaf Bop → Cat. First let us fix the notation and diagram (mostly following [Jac99]) for the cartesian lift of u at X as
u∗ (X)
u ¯(X)
/
X
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Then for a general vertical morphism f : Y1 → Y2 between two arbitrary objects in the fibre over J, we have u ¯(Y1 )
u∗ (Y1 ) u∗ (f )
u∗ (Y2 )u¯(Y I
/ Y1
2)
u
f
/ Y2 /
J
Thus every map u : I → J in B determines a functor u∗ (since each fibre can be regarded as category in its own right and therefore u∗ becomes a functor) in a reverse direction from the whole fibre EJ to the whole fibre EI . Such a functor is referred to as change-of-base or pullback functor. u∗ (X)
u ¯(X)
u∗ (f )
u∗ (g◦f )u∗ (Y
)
/
X f
u ¯(Y )
/
u∗ (g)
Y
u∗ (Z)
eI
%
I
g◦f
g
u ¯(Z)
u
/
/
Z J
x
eJ
Figure 2.9: Pullback functor from EJ to EI determined by cartesian lifts For identities and composition of arrows we have canonical natural transformations as shown in Figures 2.10, 2.11 respectively. When the cloven fibration is such that induced pullback functors make the natural transformations actual identities, then its is called a split fibration. Since split fibrations guarantee these functoriality conditions, the pseudo-functor of the cloven c 2018, Indian Institute of Technology Delhi
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idX1
XKS 1
id¯I (X1 )
α
/6 X 1
idI ∗ (X1 ) idI ∗ (X ) KS 2 α
X2
id¯I (X2 ) idX2
I
idI
/( X 2 /
I
Figure 2.10: Cartesian lift of identity in a cloven fibration u∗ v ∗ (Z)
u ¯[v ∗ (Z)]
KS
β
/ v ∗ (Z)
v¯(Z)
3/ Z
¯ (v◦u)(Z) ∗
(v ◦ u) (Z) v◦u
I
u
/
J
v
/+ K
Figure 2.11: Cartesian lift of composite arrow in a cloven fibration case actually reduces to becoming a true functor. Thus split fibrations are said to be well-behaved and comfortable to work with when such a case is possible. Fortunately most of the cases we will encounter for the purposes of signal representation would be split and would correspond to strict functors such as L2 to be discussed later. This is because from an applied perspective we mostly deal with linear spaces having single object in the fibres when they are viewed as categories. Yet it should be noted that often such equalities are not guaranteed in general for categories and the cleavages result in natural isomorphisms (instead of equalities) as discussed. For more complete exposure on this the reader is referred to [Jac99].
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2.2.6
32
B-Indexed Category and Grothendieck Construction
We now recall the definition of equivalent notion of point-wise indexing of sets in the categories which goes by the terminology of category-indexed categories. Definition 22 [Jac99] A B-indexed category is a pseudo functor Ψ : Bop → Cat. It maps each object I ∈ B to a category Ψ(I) and each morphism u : I → J ∈ B to a functor Ψ(u) : Ψ(J) → Ψ(I) with direction reversed. Such a functor Ψ(u) is denoted by u∗ , and a pseudo-functor unlike a strict functor involves natural isomorphisms αI : id u (idI )∗ and βu,v : u∗ v ∗ u (v ◦ u)∗ for usual objects I, J, K and arrows u, v in B satisfying the classic coherence conditions. Because of the natural isomorphisms for a general cloven fibration, indexed category must satisfy certain coherence conditions common in categories as shown in Figure 2.12 and Figure 2.13. u∗ αI u∗
z
(idI )∗ u∗β
u∗ αJ
$
/ u∗ o
u∗ (idJ )∗
βu,idJ
idI ,u
Figure 2.12: Coherence conditions for u : I → J
u∗ , v ∗ , w∗ βu,v w∗
(v ◦ u)∗ w∗
u∗ βv,w
/ u∗ (w ◦ v)∗
/ (w ◦ v
βv◦u,w
βu,w◦v
◦ u)∗
Figure 2.13: Coherence conditions for I →u J →v K →w L Next we recall the classic definition of Grothendieck construction. Immediately follows a Proposition 24 which will facilitate the understanding of why and how this construction turns a pseudo-functor into a fibration. But we demonstrate this only for the case of a strict functor which suffices for our requirements in an applied context. c 2018, Indian Institute of Technology Delhi
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Definition 23 [Jac99] Let Ψ : Bop → Cat be an indexed category. Then Grothendieck R Construction B Ψ of Ψ is the total category consisting of: • objects (I, X) where I ∈ B and X ∈ Ψ(I). • morphisms (I, X) → (J, Y ) are the pairs (u, f ) with u : I → J in B and f : X → u∗ (Y ) = Ψ(u)(Y ) in Ψ(I). • identity (I, X) → (I, X) is pair (id, αI (X)), with αI the natural isomorphism idΨ(I) u (idI )∗ . • composition (I, X) u (v ◦ u)∗ (Z)
(u,f )
/
(J, Y )
(v,g)
/
∗ (g)
(K, Z) where X →f u∗ (Y ) →u
u∗ v ∗ (Z)
satisfying the usual unit laws and associativity axioms with the coherence conditions guaranteeing equalities for identity and composition as shown earlier in Figures 2.12 and 2.13.
The schematic for general construction with cleavage will be denoted as (I, X1 )
(u,f1 )
/
(J, Y1 ) 6
u ¯(J,Y1 )
(id,f1 )
(I, u∗ (Y1 )) (I, u∗ (Y2 )) O
(id,f2 )
(I, X2 )
id
%
I
u ¯(J,Y2 ) (u,f2 )
u
/
(
(J, Y2 ) /
J
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(I, X) f
34
(u,f ) u ¯(Y )
6
(v,g) v¯(Z)
g
/
(K, Z)
6 9
u∗ (Y ) u∗ (g)
/ (J, Y )
u ¯[v ∗ (Z)]
6
v ∗ (Z) ¯ (v◦u)(Z)
∗ ∗
u v SK (Z)
β
(v ◦ u)∗ (Z)
eI
%
v◦u
I
u
/
J
x
eJ
v
v /+ K eK
Note that in the general cloven schematic of fibred category in the composition we have shown only second component of all objects and arrows except for the first row for simplicity to avoid clutter. One must exercise caution with notations since in the fibration definition we have made use of the notation X for an object above I in EI ; however the same notation X for an object in Ψ(I) will result into an object (I, X) above I when translated into the fibration terminology. This distinction must be kept in mind even if we slightly abuse and use the similar notations such as X in both cases of indexed categories and fibration for simplicity.
Proposition 24 [Jac99],[Vis04],[BL09] A fibred category over B with a cleavage defines a pseudo-functor Bop → Cat. Conversely from every pseudo-functor on B we get a fibred category over B with a cleavage. In-fact there exists a strict 2-equivalence between the 2-categories of pseudo-functors and cloven fibrations.
Proof : We shall not prove the proposition for the general case of a pseudo-functor but only for a strict functor since we need mostly strict functors for the work in this thesis. For this let Ψ : Bop → Cat be the strict. Now let I, J, K, .. be the objects and u, v, w, .. be the arrows of the small category B. Now the usual contra-variant
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functor diagram is given as
w
K
u◦w
/
Ψ(K) o
Ψ
− →
I
Ψ(w)
c
Ψ(I)
u
Ψ(u)
Ψ(u◦w)
J
(2.3)
O
Ψ(J)
Now to define a split fibration choose an E(I) with its second component identical (or unique up-to isomorphism) to Ψ(I) and define the objects in E to be the pairs (I, X1 ), (I, X2 ), ... where X1 , X2 , ... are objects of Ψ(I) and so on for all J, K, ... Now each functor Ψ(u) assigns a well-defined unique arrow Ψ(u)(Y1 ) to each object Y1 of Ψ(J). The claim is that these arrow will act as the cartesian lifts for the fibration and therefore one can define the arrows of this category as (u, f ) : (I, X1 ) → (J, Y1 ) where f : X1 → Ψ(u)(Y1 ). To verify this claim observe Figure 2.14, (K, Z1 ) g0
u
g
(I, Ψ(u ◦ w)(Y1 ))
h u◦w w ¯
)
(I, Ψ(u)(Y1 ))
u ¯
,/
#
(J, Y1 )
Figure 2.14: Proof for the cartesian lifts in fibration formed using a strict functor Ψ
For every arrow g to (J, Y1 ) we have to show that there exists a unique h above w as per the classic cartesian lift definition as shown in Figure 2.15. Now for every g from some (K, Z1 ) to (J, Y1 ) there is a unique g 0 : Z1 → Ψ(u ◦ w)(Y1 ) in Ψ(K) which follows from the definition of arrows in E and the fact that g is above u ◦ w. Since the functor Ψ(w) assigns a well-defined object Ψ(w)Ψ(u)(Y1 ) to the object Ψ(u)(Y1 ) and since Ψ is a true functor which implies Ψ(w ◦ u) = Ψ(w)Ψ(u) by its definition, therefore the arrow w¯ is precisely same as Ψ(w)Ψ(u)(Y1 ). Given a unique g 0 , we have a unique h = (w, g 0 ) which is the required factorization of g through u¯ proving the claim. The classic cartesian square for this case is shown in Figure 2.15. For the c 2018, Indian Institute of Technology Delhi
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(K, Z1 ) _
g h
E
'
(I, Ψ(u)(Y 1 )) _
u ¯
/+
(J,_Y1 )
P
B
K u◦w=P g w
(
I
u
,/
J
Figure 2.15: Classic fibration definition in the case of strict functor Ψ reverse case we only provide a few hints and leave the relatively easy proof for the reader. For the reverse case given a split fibration the reader may observe that the cartesian lifts of a given arrow in the base for objects in the fibers sets up a pullback functor in the reverse direction. The fibers along with these pullback functors when worked out will give rise to an associated strict contravariant functor from B to Cat. For the proof in the complex case of a pseudo-functor refer to [Jac99],[Vis04],[BL09]. The definition of fibration could be interpreted intuitively as making some part of the structure (that which is specified by the category B) within E explicit. This will become obvious after we have explored the examples and connections with usual functor characterized by the base structured categories.
2.2.7
An example of fibration, P : Vect → Fld
In this section we shall understand both intuitively and rigorously that the category of all vector spaces over arbitrary fields with linear transformations is truly a fibred category with its base as the category of all fields and field homomorphisms. Since a field acts on an Abelian group; we can define a functor from Fld to Cat which maps each object K to a full subcategory of Ab (considered as an object of c 2018, Indian Institute of Technology Delhi
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Cat) consisting of all those Abelian groups on which K can act. This functor then sends field homomorphisms to functors on these subcategories. Since the action of a field on Abelian group can be defined both as a left as well as right action we can define the above functor in a contravariant fashion giving us Fld-indexed category which is a (strict) functor Ψ : Fldop → Cat. It maps each object K ∈ Fld to the subcategory KAb consisting of Abelian groups on which K acts. Hence Ψ(K) = KAb and a morphism u : K → L is mapped to a functor Ψ(u) : LAb → KAb .
u∗ (A00 )
u ¯(A00 )
/
u∗ (f )
A00
u∗ (g◦f )u∗ (A0 )
f
u ¯(A0 )
/
u∗ (g)
A0
u∗ (A)
eK
(
K
g◦f
g
u ¯(A)
/
A /L
u
x
eL
Since functor Ψ is a strict functor it doesn’t involve natural isomorphisms and is R therefore split fibration example. Now Grothendieck construction Fld Ψ is the total category with, • objects (K, A) where K ∈ Fld and A ∈ Ψ(K). • morphisms (K, A) → (L, A0 ) are the pairs (u, f ) with u : K → L in Fld and f : A → u∗ (A0 ) = Ψ(u)(A0 ) in Ψ(K). • identity (K, A) → (K, A) is pair (idK , idA ), since idΨ(K) = (idK )∗ . • composition (K, A)
(u,f )
/
(L, A0 )
(v,g)
/
(F, A00 )
The pairs (K, A) are precisely the K-vector spaces with A, the underlying Abelian group on which the field acts as · : K × A → A. The morphisms (u, f ) with u field c 2018, Indian Institute of Technology Delhi
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(F, A0 ) _
g h
Vect
$
(K,_ A)
(u,idA )
+/
(L,_A)
P
F
Fld
u◦w=P g w
%
K
/+
u
L
Figure 2.16: Category of vector spaces as a fibration id(X,F X)
(X, F X)
(f,F f ) (g◦f,F g·F f )
(h◦g◦f,F h·F g·F f )
(W, F W )
(h,F h)
(Y, F Y ) (g,F g)
(Z, F Z)
Figure 2.17: Category (F, C, Cat) characterizing F homomorphism and f : A → A0 is an (Abelian) group homomorphism such that f (a · x) = u(a) · f (x)∀a ∈ K, x ∈ A. Thus the fiber on K is precisely the subcategory of Vect commonly denoted as VectK consisting of all K-vector spaces. The standard diagram of fibration looks something as shown in Figure 2.16 where P is the forgetful functor sending every vector space to its underlying field.
2.2.8
Connection of (F, C, Cat) to fibred category
First we illustrate the precise connection between a strict functor into Cat to the fibred category. Consider again a general functor F : C → Cat. As seen earlier this produces a category (F, C, Cat) as shown in Figure 2.17 where the image subcategory is a 2-category with categories as objects and functors as arrows. Since F X, F Y, ... are all categories and F f, F g, ... are functors it is possible to
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id(X,A1)
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(idX ,a1)
(X, A1)
(X, A2)
(idX ,a2·a1)
(idX ,a2)
(X, A3) Figure 2.18: Expansion of the node (X, F X) (X, A1)
(f,F f )|(X,A1)
/
(idX ,a1)
(idX ,a2·a1)
(X, A2)
(X, A3))
(idY ,F f (a1))
(f,F f )|(X,A2)
/
(idX ,a2)
(Y, F f (A1))
(Y, F f (A2)) (idY ,F f (a2·a1))
(idY ,F f (a2))
(f,F f )|(X,A3)
/
(Y, F f (A3))
Figure 2.19: Schematic of the category with (X, F X),(Y, F Y ) expanded. expand the second component of objects of type (X, F X) and form new objects and arrows correspondingly as shown in Figure 2.18. Observe that this structurally corresponds to a fiber of a total category in Grothendieck fibration. Next we repeat the above expansion procedure for the second component of every node in (F, C, Cat) to get a full blown structure as shown in Figure 2.19 where we only two nodes corresponding to objects X and Y are demonstrated. Here the horizontal arrows are the restrictions of functor (f, F f ) to objects. Now using Proposition 24 and its proof we can convert the associated strict functor ¯ F : Cop → Cat into an equivalent split fibration as shown in the Figure 2.20 where we have used Lemma 63 and the fact F¯ X = F X. Recall that we denote opposite arrow of f by f ◦ . Hence we have demonstrated that since F¯ is a strict functor,(F, C, Cat) can be c 2018, Indian Institute of Technology Delhi
2.3 Measure theory and functional analysis
(X, A1)
f (Y,F f (A1))
/
(idX ,a1)
(idX ,a2·a1)
(X, A2)
idX
'
f (Y,F f (A2))
/
Figure 2.20:
(Y, F f (A2)) (idY ,F f (a2·a1))
(idY ,F f (a1))
f (Y,F f (A3))
X
(Y, F f (A1))
(idY ,F f (a1))
(idX ,a2)
(X, A3))
40
/
(Y, F f (A3)) /Y
f
w
idY
F¯ fibred on C
R C
uniquely (up-to isomorphism) converted into a fibred category on C with split cleavage. The following three categories are essentially the same.
(F, C, Cat) ∼ =
Z
F¯ ∼ = X oF C
(2.4)
C
2.3
Measure theory and functional analysis
The references for this section are comprehensive volumes [Fre11], [Fre16], [Fre12], [Fre13] of the unified measure theory. We have just recalled bare minimum prerequisites and for extensive details and rigour it is highly recommended to see these volumes.
2.3.1
σ-algebra and measure space
First we will recall the definitions of σ-algebra and measure space.
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Definition 25 (111A [Fre11]) If X is a set, then a σ-algebra (or a σ-field) of subsets of X is a family ΣX ⊆ PX of subsets of X such that (i) ∅ ∈ ΣX ; (ii) for every E ∈ ΣX , its complement X \ E in X belongs to ΣX ; (iii) for every (countable) sequence hEn in∈N in ΣX , its union
S
n∈N
En belongs
to ΣX .
Definition 26 (112A [Fre11]) A measure space is a triple (X, ΣX , µ), where (i) X is a set; (ii) ΣX is a σ-algebra of subsets of X; (iii) µ : ΣX → [0, ∞] is a function such that (a) µ∅ = 0; P S (b) if hEn in∈N is any disjoint sequence in ΣX , then µ( n∈N En ) = ∞ n=0 µEn . The function µ is called a measure on X, and the members of ΣX are called measurable sets.
Now we recall concepts of negligible set and null ideal. For details refer Section 112D of [Fre11]. Let (X, ΣX , µ) be any measure space. A set N ⊆ X is negligible (or null or µ-negligible) if there exists a set E ⊆ ΣX such that N ⊆ E and µE = 0.
Proposition 27 (112D(b) [Fre11]) Let N be the family of negligible subsets of X. Then c 2018, Indian Institute of Technology Delhi
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(i) ∅ ∈ N ; (ii) if A ⊆ B ∈ N then A ∈ N ; (iii) if hAn in∈N is any sequence in N ,
S
n∈N
An ∈ N .
Proof : These are easy to prove applying the basic definition of negligible sets and properties of measure spaces; Of course µ∅ = 0. Next there exists a set E ∈ ΣX such that µE = 0 and B ⊆ E which implies A ⊆ E. Finally For each n ∈ N we choose an S En ∈ ΣX such that An ⊆ En and µEn = 0. But E = n∈N En ∈ ΣX by definition; S S S P S while n∈N An ⊆ n∈N En , and since µ( n∈N En ) ≤ ∞ n=0 µEn , so µ( n∈N En ) = 0 S again implying n∈N An ∈ N . In general a family of sets satisfying the conditions (i)-(iii) of Proposition 27 is called a σ-ideal of sets. In particular N is called the null ideal of the measure µ. Conventionally the term measurable space is used to mean a pair (X, ΣX ) where X is a set along with ΣX as a σ-algebra of its subsets. However for the purpose of this thesis we shall avoid using this terminology. Here by the phrase measurable space we mean the triple (X, ΣX , N ) where N is the null ideal of the measure µ.
Proposition 28 (211Y(a) [Fre16]) Let (X, ΣX , µ) be a measure space and for E, F ∈ ΣX write E ∼ F if E4F ∈ I where I is σ-ideal of subsets of X. We now show that ∼ is an equivalence relation on ΣX . Let B be the set of equivalence classes in ΣX for ∼; for E ∈ ΣX , write E • ∈ B for its equivalence class. We also show that there is a partial ordering ⊆ on B defined by saying that, for E, F ∈ ΣX , E • ⊆ F • ⇐⇒ E \ F ∈ I.
Proof : First we show axioms of equivalence relation hold: 1. Reflexivity E ∼ E since E4E = ∅ ∈ I. 2. Symmetry: if E4F ∈ I then F 4E ∈ I and vice versa. 3. Transitivity: if E4F ∈ I and F 4G ∈ I then using venn diagrams and the second clause that if A ⊆ B ∈ I then A ∈ I it follows that E4G ∈ I.
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Next the axioms for the partial order are easily verified. 1. Reflexivity: E • ⊆ E • ⇐⇒ E \ E = ∅ ∈ I. 2. Antisymmetry: If E • ⊆ F • ⇐⇒ E \ F ∈ I and F • ⊆ E • ⇐⇒ F \ E ∈ I then E4F ∈ I and therefore E • = F • using the third clause of ideal definition. 3. Transitivity: If E • ⊆ F • ⇐⇒ E \ F ∈ I and F • ⊆ G• ⇐⇒ F \ G ∈ I then again using venn diagrams and second clause of ideal definition, it readily follows that E • ⊆ G• ⇐⇒ E \ G ∈ I. As a special case when I = N , we have the following corollary.
Corollary 1 Let (X, ΣX , µ) be a measure space, and for E, F ∈ ΣX write E ∼ F if µ(E4F ) = 0. Show that ∼ is an equivalence relation on ΣX . Let B be the set of equivalence classes in ΣX for ∼; for E ∈ ΣX , write E • ∈ B for its equivalence class. Then there is a partial ordering ⊆ on B defined by saying that, for E, F ∈ ΣX , E • ⊆ F • ⇐⇒ µ(E \ F ) = 0.
2.3.2
Boolean algebras and measure algebras
To understand and utilize the dual relationship between the categories involving measure spaces and measure algebras, first we need to briefly review the intuition relating sets and Boolean algebras.
Definition 29 (136E [Fre11]) Let X be a set, an algebra or a field is a family A ⊆ PX of subsets of X such that (i) ∅ ∈ A; (ii) for every A ∈ E, its complement X \ A belongs to A; (iii) for every E, F ∈ A, E ∪ F ∈ A. (closed under finite unions).
Since closure under countable unions implies closure under finite unions; every σ-algebra of subsets of X is always an algebra of subsets of X. c 2018, Indian Institute of Technology Delhi
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A Boolean algebra is a Boolean ring (B, +, .) (Boolean means b2 = b for every b ∈ B) with a multiplicative identity 1B . Proposition 30 (311A(a),(b) [Fre12]) If A ⊆ PX is an algebra of subsets of any set X then (A, 4, ∩) is a Boolean algebra; its additive identity 0A is ∅ while its multiplicative identity 1A is X. Here the operations 4, ∪ are symmetric difference and union operations of set theory. Proof : We verify the axioms, which are all easily established, using Venn diagrams. 1. A4B ∈ A for all A, B ∈ A, (additive closure). 2. (A4B)4C = A4(B4C) for all A, B, C ∈ A, (additive associativity) 3. A4∅ = ∅4A = A for every A ∈ A, (additive identity) 4. A4A = ∅ for every A ∈ A, so that every element of A is its own inverse in (A, 4), and (A, 4) is a group; 5. A4B = B4A for all A, B ∈ A, (commutativity) so that (A, 4) is an abelian group; 6. A ∩ B ∈ A for all A, B ∈ A, (multiplicative closure) 7. (A ∩ B) ∩ C = A ∩ (B ∩ C) for all A, B, C ∈ A, (multiplicative associativity) 8. A ∩ (B4C) = (A ∩ B)4(A ∩ C), (A4B) ∩ C = (A ∩ C)4(B ∩ C) for all A, B, C ∈ A,(distributivity) so that (A, 4, ∩) is a ring; 9. A ∩ A = A for every A ∈ A, (Boolean axiom b2 = b) so that (A, 4, ∩) is a Boolean ring; 10. A ∩ X = X ∩ A = A for every A ∈ A, (multiplicative identity) c 2018, Indian Institute of Technology Delhi
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so that (A, 4, ∩) is a Boolean algebra and X is its identity. Thus every algebra of sets (A, 4, ∩) is isomorphic to a Boolean algebra (B, +, .) where the elements a, b, c, .. ∈ B of Boolean algebra are in bijective correspondence with sets A, B, C, ... ∈ A. While the operations satisfy a + b = A4B; a.b = A ∩ B; A \ B = a + a.b; A ∪ B = a + b + a.b. Conversely it is also true that every Boolean algebra (B, +, .) is isomorphic to some algebra of sets (A, 4, ∩). This is often known as set-theoretical version of Stone’s representation theorem. Definition 31 [nLWc] If P is a set with a partial order (P, ≤), then for any two elements u and v of P , an element w of P is defined as the meet (or infimum or greatest lower bound) of u and v, iff the following conditions are satisfied: (a) w ≤ u and w ≤ v (i.e., w is a lower bound of u and v). (b) For any z in P , such that z ≤ u and z ≤ v, we have z ≤ w (i.e., w is greater than or equal to any other lower bound of u and v). and denoted as w = u ∧ v.
Dual to meet for a partial ordered set the binary operation of join (or supremum or least upper bound) is also defined and denoted as u ∨ v. Underlying the structure of every Boolean algebra (B, +, .) is a partially ordered set (B, ≤) where the partial order is defined by a ≤ b whenever a ∨ b = b (or equivalently a ∧ b = a). Further just as in the case of partial order, we have corresponding notions of upper bounds, lower bounds, join (supremum or least upper bound), meet (infimum or greatest lower bound), (Dedekind) completeness in the case of a Boolean algebra. The binary operations join (∨) and meet (∧) on elements of algebra correspond to the set-theoretic union and intersection operations. Thus a + b + a.b = a ∨ b and a.b = a ∧ b while the lower bound is the additive identity 0B or ∅ its upper bound is the multiplicative identity 1B or X. The notion of completeness which plays fundamental role in the theory of measure algebras is briefly reviewed next. c 2018, Indian Institute of Technology Delhi
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Definition 32 (314A [Fre12]) If P is a partially ordered set then; (a) P is conditionally complete , or order-complete or Dedekind complete when every non-empty subset of P with an upper bound has a least upper bound. (b) P is σ-order-complete, or Dedekind σ-complete when (i) every countable non-empty subset of P with an upper bound has a least upper bound (ii) every countable non-empty subset of P with a lower bound has a greatest lower bound. In the special case of Boolean algebras, one half of this definition implies the other; more precisely we have, for any Boolean algebra B,
B is (Dedekind) σ-complete ⇐⇒ every non-empty countable subset of B has a least upper bound ⇐⇒ every non-empty countable subset of B has a greatest lower bound.
Note that ΣX is Dedekind σ-complete, because if hEn in∈N is any sequence in ΣX S W then n∈N En = {En : n ∈ N} is the least upper bound of {En : n ∈ N} in ΣX . Definition 33 (321A [Fre12]) A measure algebra is a pair (B, µ ¯), where (i) B is a Dedekind σ-complete Boolean algebra; (ii) µ ¯ : B → [0, ∞] is a function such that (a) µ ¯0 = 0; (b) whenever han in∈N is a disjoint sequence in B, µ ¯(supn∈N an ) =
P∞
n=0
µ ¯ an ;
(c) µ ¯a > 0 whenever a ∈ B and a 6= 0. Just as in the case of a ring (R, +, ·) with I being an ideal, the quotient set (R/I) c 2018, Indian Institute of Technology Delhi
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is also a ring (R/I, +, ·), also in the case of a Boolean algebra B with I being an ideal, the quotient ring B/I is a Boolean algebra. Refer section 312L of [Fre12] for details.
Proposition 34 (314C [Fre12]) Let B be a Dedekind σ-complete Boolean algebra and I a σ-ideal of B. Then the quotient Boolean algebra B/I is Dedekind σ-complete.
Proof : Let A ⊆ B/I be a non-empty countable set. For each u ∈ A, choose a bu ∈ B such that u = b•u . Then c = supu∈A bu is surely defined in B; consider v = c• in B/I. Now it can be shown that the map b 7→ φ(b) = b• is sequentially order-continuous. This means φ(c) = supu∈A φ(bu ) implying v = c• = supu∈A b•u = sup A. But A is arbitrary, so B/I is Dedekind σ-complete.content... For special case of Proposition 34 we have Corollary 2. Corollary 2 (314D [Fre12]) Let X be a set, ΣX a σ-algebra of subsets of X, and I a σ-ideal of subsets of X. Then ΣX ∩ I is a σ-ideal of the Boolean algebra (ΣX , 4, ∩), and ΣX /ΣX ∩ I is Dedekind σ-complete. Proof : Ofcourse if we just prove that Σ ∩ I is a σ-ideal then result automatically follows from the Proposition 34. Now ΣX ∩ I is a family of subsets of X each having a form F ∩ E where F ∈ ΣX and E ∈ I. (i) We have ∅ ∩ ∅ ∈ ΣX ∩ I satisfying first clause of σ-ideal definition. (ii) Let A ⊆ (F ∩ E) ∈ ΣX ∩ I, then A could be expressed in a form (F ∩ G) where G ⊆ E ∈ I but since G ∈ I hence A ∈ ΣX ∩ I. satisfying second clause of σ-ideal definition (iii) Let hAn in∈N be any sequence in ΣX ∩ I, where An = Fn ∩ En . Let F = S S S S ( n∈N Fn ) ∈ ΣX , then n∈N An ⊆ ( n∈N Fn ) ∩ ( n∈N En ) ∈ ΣX ∩ I. Hence by second S clause above n∈N An ∈ ΣX ∩ I satisfying third clause.
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Indeed ΣX ∩ I is a σ-ideal of the Boolean algebra ΣX , and ΣX /ΣX ∩ I is Dedekind σ-complete.
2.3.3
Atomic measure spaces and dual atomic Boolean algebras.
Definition 35 (112B [Fre11]) Suppose X is any set, and h : X → [0, ∞] is any P P function. For every E ⊆ X if we declare µE = x∈E h(x) taking x∈∅ h(x) = 0 then (X, PX, µ) becomes a measure space. Such measures µ are termed as pointsupported. P P Note that for infinite sets E one can take x∈E h(x) = sup{ x∈I h(x) : I ⊆ E is finite}, because every h(x) is non-negative. As an example, if X = N, then for P P countably infinite E it reduces to n∈E h(n) = limn→∞ m∈E,m≤n h(m). This special case of h(x) = 1 for every x, is called counting measure on X. Here µE is just the number of points of E given E is finite, else is ∞ if E is infinite. Definition 36 (211I,J,K [Fre11]) Let (X, Σ, µ) be a measure space; (i) A set A ∈ Σ is an atom for µ if µA > 0 and whenever E ∈ Σ, E ⊆ A either E or A \ E is negligible. (ii) Measure µ, or space (X, Σ, µ), is purely atomic or discrete if whenever E ∈ Σ and E is not negligible there is an atom for µ included in E. (iii)Measure µ, or space (X, Σ, µ), is atomless or diffused or continuous if there are no atoms for µ.
Every counting measure and in general point-supported measure is purely atomic because {x} must be an atom whenever µ{x} > 0. Using Boolean algebra the equivalent of Definition 36 are reviewed now. c 2018, Indian Institute of Technology Delhi
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Definition 37 (316K [Fre12]) Let B be a Boolean algebra; (i) An atom in B is a non-zero a ∈ B such that the only elements included in a are 0 and a. (ii) B is atomless if it has no atoms. (iii) B is purely atomic if every non-zero element includes an atom. A counting measure on a set X is always complete, strictly localizable and purely atomic. It is σ-finite iff X is countable, totally finite iff X is finite, a probability measure iff X is a singleton, and atomless iff X is empty.
2.3.4
Measurable, measure-preserving functions and Boolean homomorphisms
Proposition 38 (111X(c) [Fre11]) If X and Y are some sets and T is a σ-algebra of subsets of Y ; then for a function φ : X → Y , {φ−1 [F ] : F ∈ T} is a σ-algebra of subsets of X. Proof : Again we sketch the basic steps for proof. 1. A = {φ−1 [F ] : F ∈ T} 2. Y ∈ T, φ−1 [Y ] = X ∈ A 3. E ∈ T, φ−1 [E] ∈ A, (Y \ E) ∈ T, φ−1 [Y \ E] = X \ φ−1 [E] ∈ A 4. X \ φ−1 [Y ] = ∅ ∈ A 5. for every sequence hEn in∈N ∈ T its union S S S −1 −1 n∈N En ∈ T,φ [ n∈N En ] = n∈N φ [En ] ∈ A. If we associate ΣX , some σ-algebra of subsets of X along with X then a function φ : X → Y is termed as ΣX -measurable if {φ−1 [F ] : F ∈ T} ⊆ ΣX . Hence for a general real-valued function with Borel σ-algebra ΣB on the codomain R the following constitutes as one of the basic definitions of measure theory. Definition 39 (121B,C [Fre11]) Let X be any set, ΣX a σ-algebra of subsets of X, and E a subset of X. A function f : E → R is called ΣX -measurable (or measurable) if it satisfies any, or equivalently all, of following conditions c 2018, Indian Institute of Technology Delhi
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(i) {x : f (x) < a} ∈ ΣE for every a ∈ R; (ii) {x : f (x) ≤ a} ∈ ΣE for every a ∈ R; (iii) {x : f (x) > a} ∈ ΣE for every a ∈ R; (iv) {x : f (x) ≥ a} ∈ ΣE for every a ∈ R. where ΣE is the subspace σ-algebra of subsets of E ⊆ X. Of course if X is Rr , and B is its Borel σ-algebra, a Σ-measurable function is called Borel measurable. If X is Rr , and ΣL is the σ-algebra of Lebesgue measurable sets, then a ΣX -measurable function is called Lebesgue measurable. Remark 40 It is important to note that Definition 39 given in [Fre16] is generally defined on any subset E ⊆ X or in other words partial measurable functions. From the category theory perspective unless we are working with partial categories, the measurable functions will be taken as defined on total domain X. Following [Fre11], [Fre16] and [Fre12] we have retained this generality for proving theorems and propositions in thesis since we can directly use these when dealing with partial categories in measure theory. Moreover the special case of (total) measurable functions on total X is much simpler once we are used to the general case. Whenever this distinction is crucial in given context it would be explicitly mentioned. A particular one is the difference between L0 X , the space of all measurable functions from X to R and general L0 , the space of all partial functions f from E to R where E ⊆ X is conegligible and f F is measurable for some conegligible set F ⊆ X. See Definitions 53 and 54 and Remark 55 for details. By the phrase localizable measurable space we mean the triple (X, ΣX , N ) where N is the null ideal of the measure µ. Hence for this structured object the appropriate morphism intuitively is a ΣX -measurable function roughly also preserving the null structure. Definition 41 [Fre12] Let (X, ΣX , µ),(Y, ΣY , ν) be measure spaces and (X, ΣX , M), (Y, ΣX , N ) be the corresponding measurable spaces where M,N are the null ideals c 2018, Indian Institute of Technology Delhi
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of measures µ,ν respectively. A function φ : X → Y such that φ−1 [F ] ∈ ΣX for every F ∈ ΣY and µφ−1 [F ] = 0 whenever νF = 0 is called a non-singular measurable or measure-zero-reflecting function. Note that every element in the null ideal need not be measurable yet because we ensure that pre-image of every F ∈ ΣY whenever νF = 0 is also measurable with measure zero, consequently {φ−1 [N ] : N ∈ N } ⊆ M. In other words the pre-image of every element of N (σ-ideal of ν-negligible sets family) is an element of M (σ-ideal of all µ-negligible sets family). For measure spaces the appropriate morphisms roughly also preserve actual measures as defined next. Definition 42 (234A [Fre16]) If (X, ΣX , µ) and (Y, ΣY , ν) are measure spaces, a function φ : X → Y is inverse-measure-preserving if φ−1 [F ] ∈ Σ and µ(φ−1 [F ]) = νF for every F ∈ T. Definition 43 (324I [Fre12]) Let (B, µ ¯) and (A, ν¯) be measure algebras. A Boolean homomorphism φ : B → A is measure-preserving if ν¯(φb) = µ ¯b for every a ∈ B. Theorem 44 (324K [Fre12]) Let B and A be Boolean algebras and φ : B → A a Boolean homomorphism (a ring homomorphism) with kernel I. (i) If φ is sequentially order-continuous then I is a σ-ideal. (ii) If φ[B] is regularly embedded in A and I is a σ-ideal then φ is sequentially order-continuous. Proof : First note that the phrase Boolean homomorphism is simply a function φ : B → A which is a ring homomorphism. Hence by definition, φ(a4b) = φa4φb, φ(a ∩ b) = φa ∩ φb for all a, b ∈ B and φ(1B ) = 1A . Of-course the kernel of a (ring) homomorphism is the set of elements mapped to 0 and it is always an ideal in B as easily verified. 1. To prove I is a σ-ideal we just have to show it is sequentially order-closed. If han in∈N ⊆ I is a non-decreasing sequence and has a supremum c ∈ B, then φ c 2018, Indian Institute of Technology Delhi
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being sequentially order-continuous, φc = φ(supn∈N an ) = supn∈N φ(an ) = 0, so c ∈ I proving that I is a σ-ideal. 2. The term regular embedding refers to an injective order-continuous Boolean homomorphism. The sub-algebra φ[B] is regularly embedded in A means the identity map from φ[B] to A is order-continuous, implying that whenever c ∈ φ[B] is the supremum (in φ[B]) of A ⊆ φ[B], then c is also the supremum in A of A and similarly for infima. Hence it will be enough to show that φ is sequentially order-continuous when considered as a map from B to φ[B]. Suppose that han in∈N ⊆ B is non-increasing sequence and that inf n∈N an = 0. But Suppose, if possible, that 0 is not the infimum of φ[han in∈N ] in φ[B]. This means there is c ∈ B such that 0 6= φc ⊆ φan for every an ∈ han in∈N . Now φ(c \ an ) = φc \ φan = 0 for every an ∈ han in∈N , so c \ an ∈ I for every an ∈ han in∈N . The set C = {c \ an : an ∈ han in∈N } is non-decreasing and has supremum c; since I is a σ-ideal, c = sup C ∈ I, and φc = 0, contradicting c 6= 0. Thus inf n∈N φ(an ) = 0 in either φ[B] or A. But han in∈N is arbitrary, so φ is order-continuous.
Further, following corollary automatically follows since φ[B] = B/I is always regularly embedded in B/I.
Corollary 3 (313Q [Fre12]) Let B be a Boolean algebra and I an ideal of B; denote φ for the canonical map from B to B/I then φ is sequentially order-continuous iff I is a σ-ideal.
Theorem 45 (324A [Fre12]) Let (X, Σ, µ) and (Y, T, ν) be measure spaces, and ˆ for the domain of the completion (A, µ ¯), (B, ν¯) their measure algebras. Write Σ ˆ D be the subspace σµ ˆ of µ. Let D ⊆ X be a set of full outer measure, and let Σ ˆ Let φ : D → Y be a function such that φ−1 [F ] ∈ Σ ˆ D for algebra on D induced by Σ. every F ∈ T and φ−1 [F ] is µ-negligible whenever νF = 0. Then there is a sequentially order-continuous Boolean homomorphism π : B → A defined by the formula πF • = E • whenever F ∈ T, E ∈ Σ and (E ∩ D)4φ−1 [F ] is negligible.
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ˆ such that H ∩D = φ−1 [F ]; now there Proof : Let F ∈ T. Then there is an H ∈ Σ is an E ∈ Σ such that E4H is negligible, so that (E ∩ D)4φ−1 [F ] is negligible. If E1 is another member of Σ such that (E1 ∩ D)4φ−1 [F ] is negligible, then (E4E1 ) ∩ D is negligible, so is included in a negligible member G of Σ. Since (E4E1 ) \ G belongs to Σ and is disjoint from D, it is negligible; accordingly E4E1 is negligible and E • = E1• in A. What this means is that the formula offered defines a map π : B → A. It is now easy to check that π is a Boolean homomorphism, because if (E ∩ D)4φ−1 [F ], (E 0 ∩ D)4φ−1 [F 0 ] are negligible, so are ((X \ E) ∩ D)4φ−1 [Y \ F ],
((E ∪ E 0 ) ∩ D)4φ−1 [F ∪ F 0 ].
To see that π is sequentially order-continuous, let hbn in∈N be a sequence in B. For each n we may choose an Fn ∈ T such that Fn• = bn , and En ∈ Σ such that S S (En ∩ D)4φ−1 [Fn ] is negligible; now, setting F = n∈N Fn , E = n∈N En , S (E ∩ D)4φ−1 [F ] ⊆ n∈N (En ∩ D)4φ−1 [Fn ] is negligible, so π(supn∈N bn ) = π(F • ) = E • = supn∈N En• = supn∈N πbn . (Recall that the maps E 7→ E • , F 7→ F • are sequentially order-continuous, by section 321H of [Fre12].) So π is sequentially order-continuous. Now the objects considered earlier could be regarded as measurable spaces with some added structure such as null ideal or actual measure or dually as appropriate Boolean algebras. Thus the proper structure preserving maps on these objects are measurable functions which preserve this additional structure such as measurable nonsingular (measure zero-reflecting) morphisms and Inverse-measure-preserving maps and dually the corresponding Boolean homomorphisms which were reviewed in this section.
Definition 46 (214L [Fre16]) If h(Xi , Σi , µi )ii∈I is any indexed family of measure c 2018, Indian Institute of Technology Delhi
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S spaces, then by setting X = i∈I (Xi × {i}); for E ⊆ X, i ∈ I Ei = {x : (x, i) ∈ E}; we have ΣX = {E : E ⊆ X, Ei ∈ Σi for every i ∈ I}, µE =
P
i∈I
µi Ei for every E ∈ Σ.
Then (X, ΣX , µ) is a measure space and direct sum of the family h(Xi , Σi , µi )ii∈I , L denoted as i∈I (Xi , ΣXi , µXi ) The following property of direct sum suggests intuitively separating a global measurable function (or its equivalence class under =a.e. ) on (X, ΣX , µ) into local measurable functions (or classes) on subspaces (Xi , Σi , µi ) in the partition of global domain. Proposition 47 [Fre16] Let h(Xi , Σi , µi )ii∈I be a family of measure spaces, with direct sum (X, ΣX , µ). If f is a real-valued function defined on a subset of X and for each i ∈ I, if we set fi (x) = f (x, i) whenever (x, i) ∈ dom f ; the f is measurable iff fi is measurable for every i ∈ I. The next result from [Fre16] based on direct sum makes it precise that a global measurable function f or its equivalence class f • can be identified with the local measurable functions as (f φ1 , f φ2 , ...) or classes (f • φ1 , f • φ2 , ...) . Proposition 48 (241X(d) [Fre16]) If h(Xi , Σi , µi )ii∈I is a family of measure spaces, with direct sum (X, ΣX , µ). (i) Writing φi : Xi → X for the canonical maps, φi (x) = (x, i) for x ∈ Xi , it can be shown that f 7→ hf φi ii∈I is a bijection between L0 (µ) and Q 0 i∈I L (µi ). (ii) It corresponds to a bijection or a canonical isomorphism between Q L0 (µ) and i∈I L0 (µi ). (iii) This also induces an isomorphism between Lp (µ) and P Q Q p 1/p the subspace {u : u ∈ i∈I Lp (µi ), kuk = < ∞} of i∈I Lp (µi ), for i∈I ku(i)kp ) a p ∈ [1, ∞).
2.3.5
On spaces: L0 , L2 , L0 , L2
Lemma 49 (134X(g) [Fre12]) Let ΣX , ΣY be σ-algebras of subsets of X and Y respectively. Let D ⊆ X and φ : D → Y be a function such that φ−1 [F ] ∈ ΣD , for c 2018, Indian Institute of Technology Delhi
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every F ∈ T, where ΣD is the subspace σ-algebra of ΣX . For every [−∞, ∞]-real valued ΣY -measurable function g defined on C ⊆ Y , the composite function gφ is ΣX -measurable. Proof : Let A = dom gφ = φ−1 [C] and a ∈ R. Since g is ΣY -measurable, by Definition 39 there exists an F ∈ ΣY such that {y : g(y) ≤ a} = F ∩ C. On the other hand, there exists an E ∈ ΣX such that φ−1 [F ] = E ∩ D. Hence {x : gφ(x) ≤ a} = A ∩ E ∈ ΣA . Since a is arbitrary, gφ is ΣX -measurable. Theorem 50 (121E [Fre11]) Let X be any set and ΣX a σ-algebra of subsets of X. Let f and g be real-valued functions defined on domains dom f , dom g ⊆ X. (a) If f is constant then it is measurable. (b) If f and g are measurable, then f + g is measurable, where (f + g)(x) = f (x) + g(x) for x ∈ dom f ∩ dom g. (c) If f is measurable and c ∈ R a scalar, then cf is measurable, where (cf )(x) = c · f (x) for x ∈ dom f . (d) If f and g are measurable, then f × g is measurable, where (f × g)(x) = f (x) × g(x) for x ∈ dom f ∩ dom g. (e) Let hfn in∈N is a sequence of ΣX -measurable real-valued functions with domains T included in X. Let (supn∈N fn )(x) = supn∈N fn (x) for all those x ∈ n∈N dom fn for which the supremum exists in R. Then supn∈N fn is measurable. (f ) If f is measurable and h is a Borel measurable function from a subset dom h of R to R, then hf is measurable, where (hf )(x) = h(f (x)) for x ∈ dom(hf ) = {y : y ∈ dom f, f (y) ∈ dom h}. (g) If f is measurable and E ⊆ R is a Borel set, then there is an F ∈ Σ such that f −1 [E] = {x : f (x) ∈ E} is equal to F ∩ dom f . (h) If f is measurable and A is any set, then f A is measurable, where dom(f A) = A ∩ dom f and (f A)(x) = f (x) for x ∈ A ∩ dom f . c 2018, Indian Institute of Technology Delhi
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Proof : Since this is a very basic and central theorem of measure theory we sketch the technique of proof for only (a) and (b). Let ΣE denote the subspace σ-algebra of subsets of E ⊆ X. (a) Let f (x) = c, a constant for every x ∈ dom f , then {x : f (x) < a} = dom f if c < a, else ∅. Since both belong to Σdom f , f is ΣX -measurable. (b) Consider {x : f (x) + g(x) ≤ a}, but f (x) + g(x) ≤ a iff f (x) ≤ a − g(x) iff there exists a rational number q such that f (x) ≤ q ≤ a−g(x) hence {x : f (x)+g(x) ≤ S a} = q∈Q [f −1 ((−∞, q)) ∩ g −1 ((−∞, a − q))]. Since countable union of measurable sets is measurable, hence f + g is ΣX -measurable where (f + g)(x) = f (x) + g(x) for x ∈ dom f ∩ dom g. The rest of cases may be proved using similar arguments. Refer [Fre11]. Definition 51 (112D(c) [Fre12]) A set A ⊆ X is conegligible if X \ A is negligible (or in other words, there is a measurable set E ⊆ A such that µ(X \ E) = 0). Note that (i) X is conegligible (ii) if A ⊆ B ⊆ X and A is conegligible then B is conegligible T (iii) if hAn in∈N is a sequence of conegligible sets, then n∈N An is conegligible. Proposition 52 (112X(e) [Fre11]) Let (X, Σ, µ) be a measure space, and F the set of real-valued functions whose domains are conegligible subsets of X. Then (i) {(f, g) : f, g ∈ F, f ≤a.e. g} and {(f, g) : f, g ∈ F, f ≥a.e. g} are reflexive transitive relations on F, each the inverse of the other which means if f ≥a.e. g then g ≤a.e. f (ii) {(f, g) : f, g ∈ F, f =a.e. g} is their intersection, and is an equivalence relation on F.
Proof : First using Definition 51 of conegligible set recall that when f and g are realvalued functions defined on conegligible subsets of a measure space, then we write f =a.e. g, f ≤a.e. g or f ≥a.e. g to denote, respectively, f = g a.e., which means, {x : x ∈ dom(f ) ∩ dom(g), f (x) = g(x)} is conegligible, f ≤ g a.e., which means, {x : x ∈ dom(f ) ∩ dom(g), f (x) ≤ g(x)} is conegligible, f ≥ g a.e., which means, {x : x ∈ dom(f ) ∩ dom(g), f (x) ≥ g(x)} is conegligible. c 2018, Indian Institute of Technology Delhi
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Now consider {(f, g) : f, g ∈ F, f ≤a.e. g}, then it includes (f, f ) as f ≤ f a.e., since, {x : x ∈ dom(f ) ∩ dom(f ), f (x) ≤ f (x)} is surely conegligible, proving reflexivity. Also if (g, h) is included then it means g ≤ h a.e., which means, {x : x ∈ dom(g) ∩ dom(h), g(x) ≤ h(x)} is conegligible,. But this means {x : x ∈ dom(f ) ∩ dom(g) ∩ dom(h), f (x) ≤ g(x) ≤ h(x)} is conegligible,. Now using the clause if A ⊆ B ⊆ X and A is conegligible then B is conegligible, we have {x : x ∈ dom(f ) ∩ dom(h), f (x) ≤ h(x)} is conegligible, implying f ≤ h a.e. proving transitivity. Similarly once can prove the same for other relation. It is easy to observe that these are inverse of each other since {(f, g) : f, g ∈ F, f ≤a.e. g} ⇐⇒ {(g, f ) : f, g ∈ F, g ≥a.e. f }. Now their intersection consists of all real-valued functions defined on conegligible subsets such that both f ≤a.e. g and f ≥a.e. g is true which certainly means {(f, g) : f, g ∈ F, f =a.e. g}. Reflexivity and transitivity are verified as in earlier cases while symmetry newly holds now since {(f, g) : f, g ∈ F, f =a.e. g} ⇐⇒ {(g, f ) : f, g ∈ F, g =a.e. f } proving that it is an equivalence relation. A real-valued function f for which there is a conegligible set E such that the restriction of f to E, f E is measurable, is called virtually measurable. Recall D ∼ X if D4X ∈ N where N is null-ideal of subsets of X and ∼ is an equivalence relation on ΣX . But since µ(X4D) = µ(X \ D) = 0 for a measurable set D ⊆ E where E is conegligible, hence a virtually measurable function is a real-valued function whose domain is some member of the equivalence class X • .
Definition 53 (241A [Fre16]) Let (X, Σ, µ) be a measure space, L0 or L0 (µ) is the space of real-valued functions f defined on conegligible subsets of X which are virtually measurable or f E is measurable for some conegligible set E ⊆ X. Definition 54 (241Y(c) [Fre16]) Let (X, Σ, µ) be a measure space, L0X is the space of all real-valued functions f : X → R defined on X which are measurable. Remark 55 Most standard references use the same symbol for a function in L0 or L0X and for its equivalence class in L0 , however it should be noted that the space L0 is larger than the space L0X and for every f ∈ L0 there is a g ∈ L0X such that g =a.e. f . c 2018, Indian Institute of Technology Delhi
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Also whereas L0X is a Dedekind σ-complete Riesz space, L0 is not even a linear space since its members are not defined at every point of the underlying space therefore not quite measurable. More precisely their restrictions to some conegligible subsets are measurable which emphasized by using the term virtually measurable. If we write N for the subspace of L0X consisting of measurable functions that are zero almost everywhere (f = 0 a.e., means, {x : x ∈ X, f (x) = 0} is conegligible, where 0 is constant zero function on X) then the quotient space L0X /N is identical to the Dedekind σ-complete Riesz space L0 (µ), as ordered linear space. We shall evade the distinction between L0X and L0 (µ) in most arguments of this thesis since the Dedekind σ-complete Riesz space L0X parallels the Dedekind σ-complete Riesz space L0 (µ) very closely and most propositions such as Proposition 81 involving only countably many members of these spaces hold for both of them. Thus we shall deal with L0 primarily and almost all of propositions and other properties involving L0 are also valid for L0X . Proposition 56 (241B [Fre16]) If (X, Σ, µ) is a measure space, then we have the following basic properties, corresponding to Theorem 50: (a) A constant (real-valued) function defined almost everywhere in X belongs to 0
L. (b) f + g ∈ L0 for all f , g ∈ L0 (since if f F and gG are measurable, then (f + g)(F ∩ G) = (f F ) + (gG) is also measurable). (c) cf ∈ L0 for all f ∈ L0 and scalar c ∈ R. (d) f × g ∈ L0 for all f , g ∈ L0 . (e) If hfn in∈N is a sequence in L0 and f = supn∈N fn is defined almost everywhere in X, then f ∈ L0 . (f ) If f ∈ L0 and h : R → R is Borel measurable, then hf ∈ L0 (g) L0 is simply the set of real-valued functions, defined on subsets of X, which c 2018, Indian Institute of Technology Delhi
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are equal almost everywhere to some ΣX -measurable function from X to R. Hint: (i) If g : X → R is ΣX -measurable and f =a.e. g, then F = {x : x ∈ dom f, f (x) = g(x)} is conegligible therefore f F = gF is measurable (ii) If f ∈ L0 , let E ⊆ X be a conegligible set such that f E is measurable. Then A = E ∩ dom f is conegligible and f A is measurable, so there is a measurable g : X → R agreeing with f on A and g =a.e. f .
The proof for these cases can be found in Section 241B of [Fre16].
Definition 57 (241C [Fre12]) For a measure space (X, ΣX , µ), =a.e. is an equivalence relation on L0 (µ) and L0 (µ) is defined as the set of equivalence classes in L0 (µ) under =a.e. . Corresponding to f ∈ L0 (µ), its equivalence class is denoted as f • ∈ L0 (µ). Definition 58 (244A [Fre12]) For a measure space (X, ΣX , µ), and p ∈ (1, ∞), Lp = Lp (µ) is defined as the set of functions f ∈ L0 = L0 (µ) such that |f |p is integrable, and Lp = Lp (µ) is defined as the set of functions {f • : f ∈ Lp } ⊆ L0 = L0 (µ).
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Chapter 3
Base structured categories
In this chapter we introduce and study six categories forming a kind of family for which we have coined the term base structured categories. These precise mathematical expressions characterizing a functor in certain distinct ways enable us to exploit both the category and (structured) set theoretic perspectives simultaneously utilizing advantages of both these theories immensely from an application viewpoint. Although from a pure category theory perspective it can be argued that these don’t yield new data apart from that already contained in the functor except for the new perspective of treating objects as trivial categories. However we demonstrate that such a concept of looking at ordinary objects as trivial categories in-fact strengthens the possibility of applying category theory (at least partly) where traditionally objects are being treated in set-theoretic manner without realizing category theory lurking beneath the surface in applications. Moreover using these trivial categories we can form precise mathematical expressions such as X oF C where X = FX q FY q ... is the coproduct of objects considered as trivial categories which always exists in Cat even when the coproduct need not exist in the D category truly generating the possibility of characterizing the functor as multi-object action of a general category explicitly as semidirect product of purely categories. The significance of this perspective is motivated in Section 1.1. For instance being abstractly isomorphic to C (objects and the morphisms of these are in a one-to-one correspondence to each other) it seems that (F, C, D) is no more interesting than C; however as Proposition 72 proves that it is concretely different from C and therefore possibly carries additional set-theoretic structure in addition to being a category. This creates possibility of acknowledging the fact that applications where mathematical
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constructions have functorial properties can benefit from using both category theory (with well-known strength of relative perspective or analogy and comparison) and set theory (with the well-known strength of computation on elements within the objects) simultaneously through the distinct perspective of base structured categories as studied in this paper. The perspective is roughly summarized in Table 1.1. In applications such as signal representation where classical set-theoretic Hilbert and Banach theory is being used (see [HW89], for instance) ; this perspective (the graph of a functor) made us realize that categories are implicitly involved and the fundamental concept of redundancy directly affecting true information within a signal simply follows from the relative point of view as studied comprehensively in Chapter 4. Some other applications involving set-theoretic actions and symmetry also benefit from the perspective (of transformation categories) and are partially (in sense excluding manifold theory) explored in Chapter 3.
3.1 3.1.1
Functor graphs The category (F, C, D)
In set theory, a function between sets is defined as an appropriate subset of the Cartesian product set. This motivates the question if in a similar fashion could a functor be viewed as an appropriate subcategory of a product category. Given any abstract functor F : C → D using the commutative diagrams of category C and image subcategory F C, one could pair and fuse them to construct a new commutative diagram in which the usual axioms of a category are satisfied when everything is done component-wise. This is a subcategory of a product category C × D so it satisfies the axioms of category thus giving us a category characterizing the classic functor.
Definition 59 (Abstract graph of a functor) Consider a (covariant) functor F : C → D. Then (F, C, D) or GF is a category consisting of: • objects: a collection (X, F X), (Y, F Y ), ... denoted by Ob(GF ) c 2018, Indian Institute of Technology Delhi
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• morphisms: a collection GF ((X, F X), (Y, F Y )) = {(f, F f ) : (X, F X) → (Y, F Y )} • identity: for each (X, F X), the morphism 1(X,F X) = (1X , 1F X ) • composition: if (g, f ) 7→ g ◦ f in C then ((g, F g), (f, F f )) 7→ (g, F g) ◦ (f, F f ) = (g ◦ f, F g · F f ) • unit laws: for (f, F f ) we have 1(Y,F Y ) ◦ (f, F f ) = (f, F f ) = (f, F f ) ◦ 1(X,F X) • associativity: (h, F h) ◦ ((g, F g) ◦ (f, F f )) = ((h, F h) ◦ (g, F g)) ◦ (f, F f )
The schematic representation of a functor F : C → D can be denoted using commutative diagrams [Mac98] as idX
(
X
/
w
Y
0
idY
g
h◦g◦f idW
f
)
Wo
g◦f h
& w
Z
Ff
idF XF X F h◦F g◦F f
2
idZ
idF WF W
o
F g◦F f Fh
p
/
F Y idF Y Fg
'
p
F Z idF Z
Then these individual diagrams are combined in a single diagram to give a schematic representation of (F, C, D) as
id(X,F X)
(X, F X)
(h◦g◦f,F h·F g·F f )
(W, F W )
(f,F f ) (g◦f,F g·F f )
(h,F h)
(Y, F Y ) (g,F g)
(Z, F Z)
This definition should not be entirely surprising if we ponder upon the fact that every structure preserving arrow will transport a structure into a similar structure. Then pairing the structure with its image to form a new entity will naturally satisfy axioms of that structure. To us it appears to be more appropriate to call it the graph of a functor. However, since there already exists a notion of a graph of a functor (see Remarks 3.1.2 for details) in the category theory with this terminology one must exercise caution ; . Thus we have seen this category is extremely simple to c 2018, Indian Institute of Technology Delhi
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construct using the definition of a functor as a structure preserving morphism. The structure-preserving property of the morphism (functor) plays an extremely crucial role in imparting this structure (category) to the entity F : C → D. Although this construction might not yield any new data apart from the already well-studied functor, it does offer a distinct graph perspective to look at a functor which also turns out to be a fibred category utilized in formulating category-theoretic definition of redundancy in signals as briefly motivated in Section 1.1 and studied in detail in [SJd]. Next we prove that (F, C, D) is a split fibration and a split opfibration.. Definitions 17, 18 of fibration and opfibration repectively were recalled in Chapter 2. Proposition 60 Let P : (F, C, D) → C be the usual first projection functor. Then P is also a split fibration and a split opfibration. Proof : Let P : (F, C, D) → C be the first projection functor where P (X, F X) := X for all objects (X, F X) ∈ Ob((F, C, D)) and P (f, F f ) := f for all morphisms (f, F f ) ∈ mor((F, C, D)). The morphism (f, F f ) : (X, F X) → (Y, F Y ) in (F, C, D) is Cartesian over f : X → Y in C since P (f, F f ) = f and every (g, F g) : (Z, F Z) → (Y, F Y ) in (F, C, D) for which we have P (g, F g) = f h for some h : P (Z, F Z) → X, uniquely determines an (h, F h) : (Z, F Z) → (X, F X) in (F, C, D) above h with (f, F f ) ◦ (h, F h) = (g, F g).
(Z, F Z)
FZ
_
F g=F f ◦F h Fh
#
FX
Ff
/*
(g,F g) (h,F h)
FY
&
(X, _F X)
+ (f,F f )
/ (Y, F Y ) _
Z
P (Z, F Z)
g=f h h
#
f h=P (g,F f )
X
f
/*
Y
h
'
X
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f
+/
Y
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Existence is guaranteed since the arrow (g, F g) above f h = g is in one-to-one correspondence with arrow F g on the left the way we have defined our arrows in (F, C, D). Thus corresponding to a well defined F h we have (h, F h) above h. Uniqueness is also guaranteed as there is no other arrow above h because F h is the only image arrow of h guaranteed by the definition of 1-functor F proving that the arrow above f is indeed Cartesian. This argument holds for all the arrows of (F, C, D). Thus every arrow in the constructed category acts as a Cartesian lift for the underlying arrow. Thus functor P : (F, C, D) → C is indeed a fibration (or a fibred category) since for every (Y, F Y ) ∈ (F, C, D) and f : X → P (Y, F Y ) in C, there is a Cartesian morphism (f, F f ) : (X, F X) → (Y, F Y ) in (F, C, D) above f . It is split since the cleavage satisfies the standard splitting conditions γ(idY , (Y, F Y )) = id(Y,F Y ) and γ(g, (Z, F Z)) ◦ γ(f, (Y, F Y )) = γ(g ◦ f, (Z, F Z)) both of which are a consequence of the fact that F is not pseudo but rather a strict functor. In a similar fashion it can be proved that P : (F, C, D) → C is also a split opfibration using the fact that for every (X, F X) ∈ (F, C, D) and f : P (Y, F Y ) → Y in C, there is a opCartesian morphism (f, F f ) : (X, F X) → (Y, F Y ) in (F, C, D) above f . Thus when (F, C, D) is viewed as fibred on C it appears as shown in Figure 3.1. Once we make the choice of pullbacks (note in this case the pullbacks are not just (g◦f,F gF f )
(X, FO X) (idX ,idF X )
(X, F X)
idX
'
/
(Y, FO Y )
(f,F f )
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(Z, FO Z)
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f
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X
(g,F g)
/
Y
w
idY
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g
/
(Z, F Z) w /, Z idZ
Figure 3.1: (F, C, D) with trivial categories as fibres, on C unique up-to vertical isomorphism but actually unique as there is single object in each fibre) we can observe that indeed P : (F, C, D) → C becomes a split fibration. Here every map f : X → Y in C determines a functor f ∗ in a reverse direcc 2018, Indian Institute of Technology Delhi
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tion from the whole fibre on Y to the whole fibre on X. Such a functor in fibration theory is referred to as change-of-base or pullback functor. This fibration viewf¯(Y,F Y )=(f,F f )
/
f ∗ (Y, F Y ) = (X, F X)
(Y, F Y )
f ∗ (id(Y,F Y ) )
id(Y,F Y )
/
f ∗ (Y, F Y ) = (X, F X)¯
f (Y,F Y )=(f,F f )
X
f
(Y, F Y )
/
Y
Figure 3.2: Pullback functor f ∗ corresponding to f in fibration P : (F, C, D) → C point crucially implies thinking of ordinary objects along with their identity as trivial categories which is explained in Section 3.2. However the Definition 59 of abstract graph of a functor motivates us to characterize a related functor F : C → D → Set as its concrete version utilizing 2-category nature of Set.
Definition 61 (Concrete graph of a functor) Consider a (covariant) functor F : C → D with (D, U ) being a concrete category over Set or a faithful U : D → Set with F = U ◦ F . Then (F, C, Set) is a category consisting of: • objects: the pairs (X, x) where X ∈ Ob(C) and x ∈ FX = U (F X) • morphisms: pairs (f, Ff |x ) : (X, x) → (Y, y) where f : X → Y ∈ C, y = Ff (x) • identity: for (X, x), the morphism id(X,x) = (idX , idFX |x ) • composition: (g, Fg|y ) • (f, Ff |x ) = (g ◦ f, F(g ◦ f )|x ) • unit laws: for (f, Ff |x ), (idY , idFY |y ) • (f, Ff |x ) = (f, Ff |x ) = (f, Ff |x ) • (idX , idFX |x ) • associativity: (h, Fh|z )•((g, Ff |y )•(f, Ff |x )) = ((h, Fh|z )•(g, Ff |y ))•(f, Ff |x ) = (h, Fh|z ) • (g, Ff |y ) • (f, Ff |x ) c 2018, Indian Institute of Technology Delhi
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This category will now be proved only as opfibration in contrast to the dual nature of P : (F, C, D) → C as both fibration and opfibration.
Proposition 62 Let P : (F, C, Set) → C be the usual first projection functor. Then P is a split opfibration.
Proof : Let P : (F, C, Set) → C be the first projection functor where P (X, x) := X for all objects (X, x) ∈ Ob((F, C, Set)) and P (f, Ff |x ) := f for all morphisms (f, Ff |x ) ∈ mor((F, C, Set)). The morphism (f, Ff |x ) : (X, x) → (Y, y) in (F, C, Set) is opCartesian over f : X → Y in C since P (f, Ff |x ) = f and every (g, Fg|x ) : (X, x) → (Z, z) in (F, C, Set) for which we have P (g, Fg|x ) = hf for some h : Y → P (Z, z), uniquely determines an (h, Fh|y ) : (Y, y) → (Z, z) in (F, C, Set) above h with (h, Ff |y ) ◦ (f, Ff |x ) = (g, Fg|x ).
>4
(Fg|x )
x
(g,Fg|x )
/
Ff |x
(Fh|y )
y
f
(h,Fh|y )
(X, x) _
(f,Ff |x )
/
(Y, y) _
3 P9 (Z, z)
5? Z
h◦f =g
X
z) 3 (Z, : _
z
/Y
h◦f =P (g,Fg|x )
h
X
f
/
h
Y
The existence is guaranteed since the arrow (g, Fg|x ) above hf = g is in one-toone correspondence with arrow Fg|x on the left where Fg|x = F(h ◦ f )|x using the fact that F is a usual functor. Thus corresponding to a well defined Fh|y we have (h, Fh|y ) above h. The uniqueness is guaranteed as there is no other arrow above h with domain as (Y, y) since Fh|y is the unique restriction of function or functor Fh to element or object y, proving that the arrow above f is indeed opCartesian. This argument holds for all the arrows of (F, C, Set). Thus every arrow in the constructed category acts as a opCartesian lift for the underlying arrow. c 2018, Indian Institute of Technology Delhi
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Thus functor P : (F, C, Set) → C is indeed an opfibration since for every (X, x) ∈ (F, C, Set) and f : P (X, x) → Y in C, there is a opCartesian morphism (f, Ff |x ) : (X, x) → (Y, y) in (F, C, Set) above f . It is split since the opcleavage satisfies the splitting conditions κ(idX , (X, x)) = id(X,x) and κ(g, (Y, y)) ◦ κ(f, (X, x)) = κ(g ◦ f ), (X, x)) both of which are a consequence of the fact that F is not just pseudo but a strict 2-functor where Set is treated as a 2-category.
3.1.2
Remarks on relationship of (F, C, D) to Graph(F)
As mentioned earlier, there is an existing notion of graph of a functor [nLWa] within category theory denoted as Graph(F ). The existing graph of a functor is defined using the notions of profunctor and subobject classifier in a topos theoretic way. For a succinct and intuitive overview of topos theory; see [Lei10] and references therein. More precisely, the graph of F is the fibration Graph(F ) → Cop × D classified by χF . Using this definition [nLWa] one can easily recover the ordinary notion of graph of a function as a subset of set X × Y ; however the notion of graph of any structure preserving map is itself an object with that structure is not recoverable by this notion since like Set every category need not have a sub-object classifier. Indeed (F, C, D) as a category the way we defined cannot be recovered since Cat is not a topos. Hence if the reader wishes to interpret (F, C, D) as the graph of a functor then appropriate care needs to be taken.
3.2
Transformation categories
In Section 3.1.1 we saw that (F, C, D) was constructed as a graph of a functor and shown as fibred category. This motivates us to seek a corresponding contravari¯ : Cop → Cat whose ant functor Ψ : C → Cat thought of as covariant functor Ψ Grothendieck construction must yield a fibred category at least equivalent to (F, C, D) or possibly its opposite. This corresponding functor as it turns out is precisely the ¯ : Cop → Cat which is constructed from F¯ : Cop → D by contravariant functor F trivial categorification of D objects or more precisely by post composing with trivial c 2018, Indian Institute of Technology Delhi
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¯ = I ◦ F¯ as explained in this section. inclusion functor I to form F
3.2.1
Trivial categorification of D objects
To be able do Grothendieck construction on ordinary functor F : C → D we need a functor I : D → Cat. The recipe for definition of this functor is intuitively suggested by Figure 3.2 using classic pullback interpretation of a fibration. As shown then in Figure 3.3 the object F X is mapped to a category I(F X) which consists of a single object F X with its identity arrow idF X . Then every morphism F f : F X → F Y of the category D is naturally mapped to a functor I(F f ). The commutative diagram is the unit law in D. This we term trivial categorification in the sense that every object is treated as trivial category while every arrow correspondingly gives rise to a functor. Note that this can be done unambiguously for every object of any arbitrary category since for every object F X the identity arrow idF X uniquely exists setting up a definition of trivial category FX = I(F X). Next for every arrow F f : F X → F Y the well-defined domain and codomain sets up a definition for a functor I(F f ) = Ff : FX → FY (one can easily verify functor axioms) sending the domain trivial category to codomain trivial category and the corresponding commutative diagram always holds as a unit law in the original category D. Note that I indeed satisfies axioms of a functor. FX
Ff
/
I(F f )(F X) = F Y I(F f )(idF X )=idF Y
idF X
FX
FX
Ff
Ff
/
I(F f )(F X) = F Y
/
FY
Figure 3.3: Objects F X, F Y mapped to trivial categories and morphisms F f mapped to functors by I. Thus corresponding to F : C → D, we have F : C → Cat as a well defined functor with F = I ◦ F . The terminology categorification is in line with its wellc 2018, Indian Institute of Technology Delhi
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studied usage [JCB98] in context of replacing sets with categories and functions with functors, since here we are replacing objects (either abstract or concrete such as Hilbert spaces, smooth manifolds etc.) with categories while morphisms are replaced with functors.
3.2.2
R
Right action induced by a functor:
¯ Cop F,
R Cop
¯ F
The basic duality fact in [Mac98] states that every contravariant functor could be written as covariant using the concept of opposite category. This can be utilized in a reverse way to express a covariant functor as a contravariant functor. The contravariant form is essential since the original version of Grothendieck construction [Gro71] is on a contravariant pseudofunctor and the same version is utilized by [Jac99] in the context of categorical logic. As in [Gro71] let us denote dual arrows f op by f ◦ in diagrams for convenience.
Lemma 63 [Mac98] To every (covariant) functor F : C → D we can always associate a corresponding (contravariant) functor F¯ : Cop → D.
Proof : Consider the functor F : C → D. By definition it assigns to each object X ∈ C an object F X ∈ D and to each arrow f : X → Y ∈ C an arrow F f : F X → F Y ∈ D with F (g ◦ f ) = (F g) • (F f ) whenever g ◦ f is defined. Now we write F¯ f op for F f and F¯ X = F X ; then one can define a functor F¯ which is contravariant from Cop to D assigning to each object X ∈ Cop an object F¯ X ∈ D and to each arrow f op : Y → X an arrow F¯ f op : F¯ X → F¯ Y (in the opposite direction) all in such a way that F¯ (idX ) = idF¯ X and F¯ (f op ◦ g op ) = (F¯ g op ) • (F¯ f op ) whenever the composite f op ◦ g op is defined in Cop . Thus the contravariant functor inverts the order of composition as explained in [Mac98]. FX O
Ff
F
X
/ FY O
Fg
F f
/
Y
/
F OZ F
g
/
Z
¯
¯
◦
◦
F f =F f / F¯ Y = F Y F g=F g / F¯ Z = F Z F¯ X =O F X O O F¯
F¯
Xo
f◦
Y o
F¯ g◦
c 2018, Indian Institute of Technology Delhi
Z
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The functor F¯ : Cop → D as we have defined is only symbolic on objects of the original category and in general is undefined on C. Only in the case of categories such as groups, groupoids, partial monic (which are in essence partial groupoids) etc; the opposite category is isomorphic to original category i.e Cop ∼ = C. In such a case we can meaningfully write a contravariant functor F¯ : C → D and therefore the R ¯ could be obtained using the Grothendieck construction. More often is category C F the case of equivalence Cop ∼ = E where opposite category is equivalent to some other category E rather than original C. Lemma 63 together with F = I ◦ F immediately suggests Grothendieck construction in case of ordinary contravariant functors (to ordinary category D instead of Cat). Given an ordinary covariant functor F : C → D, we first think of it as a contravariant functor F¯ : Cop → D. This lets us define Grothendieck construction on R ¯ : Cop → Cat or more precisely the category op F ¯ where F ¯ = I ◦ F¯ . it by using F C This is in disguise left action of Cop which through duality can be interpreted as right action of C. However unless C is isomorphic to its opposite Cop the right action and left action categories are in general not isomorphic; see Section 3.2.4. Definition 64 (Abstract Right action induced by a functor) Consider a strict ¯ : Cop → contravariant functor F¯ : Cop → D between small categories thought of as F ¯ = I ◦ F¯ and I : D → Cat as defined). Then Grothendieck construction Cat (with F R ¯ is a category op F ¯ with of F C • objects: the pairs (X, F¯ X) where X ∈ Ob(Cop ) and F¯ X ∈ Ob(D) R ¯ • morphisms: Cop F((Y, F¯ Y ), (X, F¯ X)) are pairs (f ◦ , idF¯ Y ) where f ◦ : Y → op ¯ ◦ (F¯ X) X ∈ C idF¯ Y : F¯ Y → Ff • identity: for (X, F¯ X), the morphism id(X,F¯ X) = (idX , idF¯ X ) ¯ ◦ (idF¯ Y ) · idF¯ Z ) = (f ◦ g ◦ , idF¯ Z ) • composition: (f ◦ , idF¯ Y ) • (g ◦ , idF¯ Z ) = (f ◦ ◦ g ◦ , Fg ¯ ◦ (idF¯ Y ) : Fg ¯ ◦ F¯ Y → Fg ¯ ◦ Ff ¯ ◦ (F¯ X) since Fg • unit laws: for (f ◦ , idF¯ Y ), (idX , idF¯ X ) • (f ◦ , idF¯ Y ) = (f ◦ , idF¯ Y ) = (f ◦ , idF¯ Y ) • (idY , idF¯ Y ) c 2018, Indian Institute of Technology Delhi
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• associativity: (f ◦ , idF¯ Y ) • ((g ◦ , idF¯ Z ) • (h◦ , idF¯ W )) = ((f ◦ , idF¯ Y ) • (g ◦ , idF¯ Z )) • (h◦ , idF¯ W ) = (f ◦ , idF¯ Y ) • (g ◦ , idF¯ Z ) • (h◦ , idF¯ W ) /
(X, FO¯ X) l (idX ,idF¯ X )
(X, F¯ X) O
(idX ,idF¯ X )
(idZ ,idF¯ Z )
i
(Y, F¯ Y )
F¯ f ◦ (f ◦ ,idF¯ Y )
(X, F¯ X) '
¯ ◦ Ff ¯ ◦ (F¯ X)) (Z, Fg O
(f ◦ ◦g ◦ ,idF¯ Z )
(X, F¯ X)
idX
/
¯ ◦ (F¯ X)) (Y, Ff
X ol
/
(Z, F¯ Z)
¯ ◦ (F¯ X)) (Y, Ff O
j
(idY ,idF¯ Y )
F¯ g ◦
¯ ◦ (F¯ Y )) (Z, Fg O
(idZ ,idF¯ Z )
(g ◦ ,idF¯ Z )
(Y, F¯ Y )
f◦
/
w Y o
idY
(Z, F¯ Z)
g◦
Z
w
idZ
f ◦ ◦g ◦
Figure 3.4:
R Cop
¯ fibred on Cop ; dotted arrows show D morphisms as actions F
The abstract right action immediately motivates us to define a concrete version given there is some underlying category of D. We define this for Set (or a construct) however the case for any underlying category X should not be more difficult.
Definition 65 (Concrete Right action induced by a functor) Consider a strict contravariant functor F¯ : Cop → D between small categories with (D, U ) being a con¯ = U ◦ F¯ and crete category over Set or a faithful functor U : D → Set. Then F R ¯ is a category with F Cop ¯ = U (F¯ X) • objects: the pairs (X, x) where X ∈ Ob(Cop ) and x ∈ FX • morphisms: pairs (f ◦ , y) : (Y, y) → (X, x) where f ◦ : Y → X ∈ Cop , x = ¯ (y) Ff • identity: for (X, x), the morphism id(X,x) = (idX , x) ¯ ◦ Ff ¯ ◦ (x) • composition: (f ◦ , y) • (g ◦ , z) = (f ◦ ◦ g ◦ , z) since z = Fg c 2018, Indian Institute of Technology Delhi
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• unit laws: for (f ◦ , y), (idX , x) • (f ◦ , y) = (f ◦ , y) = (f ◦ , y) • (idY , y) • associativity: (h◦ , x) • ((f ◦ , y) • (g ◦ , z)) = ((h◦ , x) • (f ◦ , y)) • (g ◦ , z) = (h◦ , x) • (f ◦ , y) • (g ◦ , z) ¯ ◦ (x)) / (Y, Ff
(X, x) l O
(idX ,x)
(X, x) i O
(idX ,x)
(Y, F¯ Y )
U (F¯ f ◦ ) (f ◦ ,y)
(X, x) '
(idZ ,z)
(f ◦ ◦g ◦ ,z)
(X, x)
idX
¯ ◦ Ff ¯ ◦ (x)) / (Z, Fg O
X lo
/
(Z, F¯ Z)
¯ ◦ (F¯ X)) (Y, Ff O
j
(idY ,y)
U (F¯ g ◦ )
¯ ◦ (F¯ Y )) (Z, Fg O
(idZ ,z)
(g ◦ ,z)
(Y, y)
f◦
/
w Y o
idY
(Z, z)
g◦
Z
w
idZ
f ◦ ◦g ◦
Figure 3.5:
¯ fibred on Cop ; dotted arrows denote concrete functions as right acF tions
R
C
In a dual sense we define the abstract and concrete versions of left actions induced by a functor and then proceed to describe this duality in Section 3.2.4.
3.2.3
Left action induced by a functor: X oF C , X oF C
Definition 66 (Abstract Left action induced by a functor) Consider a (covariant) functor F : C → D between small categories thought of as covariant F : C → R ¯ op ) is Cat (with F = I ◦ F and I : D → Cat as defined). Then X oF C (or ( Cop F) a category with • objects: the pairs (X, F X) where X ∈ Ob(C) and F X ∈ Ob(D) • morphisms: X oF C((X, F X), (Y, F Y )) are pairs (f, idF Y ) where f : X → Y ∈ C, idF Y : Ff (F X) → F Y c 2018, Indian Institute of Technology Delhi
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• identity: for (X, F X), the morphism id(X,F X) = (idX , idF X ) • composition: (g, idF Z ) • (f, idF Y ) = (g ◦ f, idF Z · Fg(idF Y )) = (gf, idF Z ) since Fg(idF Y ) = FgFf (F X) → Fg(F Y ) • unit laws: for (f, idF Y ), (idY , idF Y ) • (f, idF Y ) = (f, idF Y ) = (f, idF Y ) • (idX , idF X ) • associativity: (h, idF W ) • ((g, idF Z ) • (f, idF Y )) = ((h, idF W ) • (g, idF Z )) • (f, idF Y ) = (h, idF W ) • (g, idF Z ) • (f, idF Y )
(X, FO X)
(g◦f,idF Z )
(idX ,idF X )
/
(X, F X)
(X, FO X) (idX ,idF X )
(X, F X)
idX
'
2
(Y, F Y )
(f,idF Y ) Ff
(idZ ,idF Z )
/
(Y, Ff (F X))
5
(Y, FO Y )
(idY ,idF Y )
/
(Y, Ff (F X))
(Z, FO Z)
(Z, FgFf (F X))
Z) 4 (Z, F O
(f,idF Z ) Fg
/
(idZ ,idF Z )
(Z, Fg(F Y ))
g◦f
X
f
/
Y
w
idY
g
w /, Z idZ
Figure 3.6: X oF C fibred on C; dotted arrows show D morphisms as actions
The abstract left action again motivates us to define a concrete version given there is some underlying category of D. We define this for Set (commonly known as construct) however it should work for any underlying category X similarly.
Definition 67 (Concrete Left action induced by a functor) Consider a covariant functor F : C → D between small categories with (D, U ) being a concrete category R ¯ op ) is over Set or a faithful U : D → Set. Then F = U ◦ F and X oF C (or ( Cop F) a category with • objects: the pairs (X, x) where X ∈ Ob(C) and x ∈ FX = U (F X) c 2018, Indian Institute of Technology Delhi
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• morphisms: pairs (f, y) : (X, x) → (Y, y) where f : X → Y ∈ C, y = Ff (x) • identity: for (X, x), the morphism id(X,x) = (idX , x) • composition: (g, z) • (f, y) = (g ◦ f, z · Fg(y)) = (g ◦ f, z) since z = FgFf (x) • unit laws: for (f, y), (idY , y) • (f, y) = (f, y) = (f, y) • (idX , x) • associativity: (h, w) • ((g, z) • (f, y)) = ((h, w) • (g, z)) • (f, y) = (h, w) • (g, z) • (f, y)
(X, x)
(g◦f,z)
(idX ,x)
/
(X, x)
(X, x) O
(idX ,x)
(X, x)
idX
'
2 (Z, z) O
(Y, y)
O
(f,y) U (F f )
/
(Y, Ff (x))
6
(Y, y) O
(g,z)
(idY ,y)
/
(idZ ,z)
(Y, Ff (x))
U (F g)
g◦f
X
f
/
Y
w
idY
g
(Z, FgFf (x)) 5 (Z,O z) (idZ ,z)
/ (Z, Fg(y)) w /, Z idZ
Figure 3.7: X oF C fibred on C; dotted arrows denote concrete functions as left actions
3.2.4
Duality between the categories defined as right actions and left actions
R ¯ is identical to Now observing carefully we can discern that the opposite of Cop F R ¯ op = X oF C. For this note that we first make use X oF C or precisely ( Cop F) R ¯ op are of the fact that F¯ X = F X and then (Cop )op = C. Thus objects in ( Cop F) the pairs (X, F¯ X) which are same as (X, F X) of X oF C. Next consider the arrow R R ¯ The opposite arrow of this in ( op F) ¯ op (f ◦ , idF¯ Y ) : (Y, F¯ Y ) → (X, F¯ X) of Cop F. C is (f, idF Y ) : (X, F X) → (Y, F Y ) which is same as the arrow of X oF C. Indeed the reader can verify that these categories are identical. c 2018, Indian Institute of Technology Delhi
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In the special case of groups, groupoids, partial groupoids (partial monic) categories where arrows can be inverted uniquely we have Cop ∼ = C via the inverse map on arrows and therefore both these conventions of right and left actions are the same. R R ¯ respectively ¯ and In these cases the abstract and concrete actions reduce to C F F C as defined next. These are isomorphic to left action categories. Definition 68 (Abstract Right action induced by a functor when Cop ∼ = C) Consider a strict contravariant functor F¯ : C → D between small categories thought R ¯ : C → Cat (with F ¯ = I ◦ F¯ and I : D → Cat as defined). Then ¯ is a of as F F C category with • objects: the pairs (X, F¯ X) where X ∈ Ob(C) and F¯ X ∈ Ob(D) R ¯ • morphisms: C F((X, F¯ X), (Y, F¯ Y )) are pairs (f, idF¯ X ) where f : X → Y ∈ ¯ ¯ ¯ C, idF¯ X : F X → Ff (F Y ) • identity: for (X, F¯ X), the morphism id(X,F¯ X) = (idX , idF¯ X ) ¯ (idF¯ Y ) · idF¯ X ) = (gf, idF¯ X ) since • composition: (g, idF¯ Y ) • (f, idF¯ X ) = (g ◦ f, Ff ¯ (idF¯ Y ) : Ff ¯ F¯ Y → Ff ¯ Fg( ¯ F¯ Z) Ff • unit laws: for (f, idF¯ X ), (idY , idF¯ Y ) • (f, idF¯ X ) = (f, idF¯ X ) = (f, idF¯ X ) • (idX , idF¯ X ) • associativity: (h, idF¯ Z ) • ((g, idF¯ Y ) • (f, idF¯ X )) = ((h, idF¯ Z ) • (g, idF¯ Y )) • (f, idF¯ X ) = (h, idF¯ Z ) • (g, idF¯ Y ) • (f, idF¯ X )
Definition 69 (Concrete Right action induced by a functor when Cop ∼ = C) Consider a strict contravariant functor F¯ : C → D between small categories with ¯ = U ◦ F¯ (D, U ) being a concrete category over Set or a faithful U : D → Set. Then F R ¯ is a category with and C F ¯ = U (F¯ X) • objects: the pairs (X, x) where X ∈ Ob(C) and x ∈ FX ¯ (y) • morphisms: pairs (f, x) : (X, x) → (Y, y) where f : X → Y ∈ C, x = Ff • identity: for (X, x), the morphism id(X,x) = (idX , x) c 2018, Indian Institute of Technology Delhi
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¯ Fg( ¯ F¯ Z)) o (X, Ff
¯ F¯ Z)) o (Y, Fg(
O
(g◦f,idF¯ X )
(idX ,idF¯ X )
(X, F¯ X) ¯ (F¯ Y )) o (X, Ff O (idX ,idF¯ X )
idX
F¯ f (f,idF¯ X )
Figure 3.8:
R C
(Z, F¯ Z)
¯ F¯ Z)) o (Y, Fg( O 4
F¯ g
(g,idF¯ Y )
(idY ,idF¯ Y )
(Y, F¯ Y ) g◦f
X
(idZ ,idF¯ Z )
(Y, F¯ Y )
(X, F¯ X) '
¯ Z) 2 (Z, F O
f
/
Y
w
idY
5
(Z, FO¯ Z) (idZ ,idF¯ Z )
(Z, F¯ Z)
g
w /, Z idZ
¯ fibred on C ∼ F = Cop ; dotted arrows show D morphisms as actions
¯ Fg(z) ¯ • composition: (g, y) • (f, x) = (g ◦ f, x) since x = Ff • unit laws: for (f, x), (idY , y) • (f, x) = (f, x) = (f, x) • (idX , x) • associativity: (h, z) • ((g, y) • (f, x)) = ((h, z) • (g, y)) • (f, x) = (h, z) • (g, y) • (f, x) R R R R ¯ X oF C (or ( op F) ¯ op ) defined ¯ op F ¯ op ) X oF C (or ( op F) The categories Cop F, C C C and visualized in this section generalize the concept of monoid action to category action. The generalization is in sense that there are simultaneous actions on multiple objects unlike a single object in a monoid action. More accurately in category action the object X (on which F or F defines an action) is a coproduct object inside the category Cat or Set respectively. This denotes actually the family of all the objects which lie in the image subcategory F (C) and F(C) each treated as a category either trivially or usual categorification of its underlying set. More precisely, X = qX∈Ob(C) F(X) X = qX∈Ob(C) F(X)
(3.1)
Symbolically this is same as C × X → X . Of course the individual objects F X in the category F (C) are acted upon by all the arrows f : X → Y of C whose domain or source object is X which can be captured by defining action set-theoretically or c 2018, Indian Institute of Technology Delhi
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77
¯ Fg(z)) ¯ o (X, Ff
¯ o (Y, Fg(z))
O
(idX ,x)
(X, x) ¯ (y)) o (X, Ff O
(idX ,x)
idX
Figure 3.9:
(Z, z) O
U (F¯ f ) (f,x)
5
¯ o (Y, Fg(z)) O
(Z, z) U (F¯ g) (g,y)
(idY ,y)
(Y, y) g◦f
X
(idZ ,z)
(Y, y)
(X, x) '
2
(g◦f,x)
f
/
Y
w
idY
6
(Z, z) O
(idZ ,z)
(Z, z)
g
,/
Z
w
idZ
¯ fibred on C ∼ F = Cop ; dotted arrows denote concrete functions as right actions R
C
element wise. Indeed using standard functor definition we can observe that every f defines a corresponding well-defined arrow F f : F X → F Y in the category D. The action perspective in the context of symmetry will be revisited in the Chapter 5 of this thesis. However we briefly state and prove that the usual transformation groupoid X//G can be viewed as a base structured category. It is well-known in the groupoid literature [Bro87],[Wei96] that a general group action gives rise to a transformation groupoid. More precisely,
Definition 70 [JCM14] Let φ : G → Aut(X), be a usual (set-theoretic) group action then transformation groupoid X//G is the groupoid consisting of: • Objects: each element x ∈ X; denoted as Ob(X//G) = X • Morphisms: (X//G)(x, y) = (g, x) ∈ G × X, with φg (x) = y • Composition: (g 0 , φg (x)) ◦ (g, x) = (g 0 g, x)
Now we show that every transformation groupoid is also a base structured category. c 2018, Indian Institute of Technology Delhi
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Proposition 71 A classic transformation groupoid X//G is isomorphic to a baseR ¯ where C = G the group G treated as one object category. structured category Cop F Proof : The group G can be viewed as a single object category G with an object ? and G(?, ?) = G. But since Gop ∼ = G we directly use the definition 69 where we consider a strict contravariant functor F¯ : G → Set between small categories with (Set, id) being modeled as a concrete category over itself or an underlying functor R ¯ = id ◦ F¯ = F¯ and U = id : Set → Set. Then F F¯ becomes a category with G objects the pairs (?, x) where ? ∈ Ob(G) and x ∈ F¯ (?) = X, morphisms the pairs (g, x) : (?, x) → (?, y) where g : ? → ? ∈ G, x = F¯ g(y). The composition is given by (g 0 , y) • (g, x) = (g 0 ◦ g, x) since x = F¯ g F¯ g 0 (z). Finally comparing with the classic R X//G, we find that the objects of X//G are simply the objects of F¯ relabeled by G
dropping the first component while the morphisms are identical. Indeed we have R X//G ∼ = G F¯ and in fact the transformation groupoid is just the special case of the R ¯ general base structured category Cop F. We now state and prove a proposition interrelating the base structured categories. Proposition 72 Let F : C → D be any ordinary abstract 1-functor (thought of as F I a functor F : C − →D→ − Cat as defined earlier). Then the following categories are abstractly isomorphic, C∼ (3.2) = (F, C, D) ∼ = X oF C In addition if Cop is isomorphic to C then following categories are abstractly isomorphic Z ¯ C∼ F (3.3) = (F, C, D) ∼ = X oF C ∼ = C
Further additionally if (D, U ) is a concrete category over Set then the following basestructured categories become concretely isomorphic, Z ¯∼ (F, C, Set) ∼ F (3.4) = = X oF C C
Being subcategories of C × D or C × Set the base-structured categories have a first projection functor (which is restriction of the usual first projection functor) onto C. c 2018, Indian Institute of Technology Delhi
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R ¯ to C are the classic split The first projection functors sending (F, C, D), X oF C, C F R ¯ X oF C to C (op)fibrations. The first projection functors sending (F, C, Set), C F, are the classic split opfibrations. Proof : First consider the category (F, C, D).This is a well-defined category with objects the pairs (X, F X), (Y, F Y ), ... from C and F C and morphisms F((X, F X), (Y, F Y )) = {(f, F f ) : (X, F X) → (Y, F Y )} considered pairwise from C and F C. It is easy to observe that it is abstractly isomorphic to C by noting both the objects and the morphisms of each of these categories standing in a one-to-one correspondence to each other. In other words they are bijective on objects and on morphism sets. Same holds for X oF C the way we have defined this category via Grothendieck construction or R ¯ op and consequently we have precisely as ( Cop F) C∼ = (F, C, D) ∼ = X oF C
(3.5)
Next if Cop is isomorphic to C, we have a well-defined contravariant functor F¯ : ¯ : C → Cat (with C → D between small categories which could be thought of as F R ¯ = I ◦ F¯ and I : D → Cat as defined). Then ¯ is a category as defined earlier F F C in the sense of abstract right category action. Abstract isomorphism with C is easy to see and we have, Z ¯ C∼ F (3.6) = (F, C, D) ∼ = X oF C ∼ = C
It is crucial to note that the base-structured categories cannot be made concretely isomorphic to the base category C under any circumstances (such as if C and D are taken to be concrete and F is made concrete) since there are additional components in the base-structured categories forgotten by the usual (first) projection functor. These are not bijective at the level of elements of underlying sets and consequently there is additional structure at level of sets which remains transparent to category theory supporting the intuition of trivial categorification (the objects which could be structured sets are treated simply as trivial categories concealing the structure from category theory). Next let (D, U ) be concrete category with some underlying category X. Then we c 2018, Indian Institute of Technology Delhi
3.2 Transformation categories
(X, x)
80
O
(idX ,x)
(X, x)
6
(f,y)
¯ (y)) (X, Ff
(Y, y) O
O
(idY ,y)
(f,Ff |x )
/
(idX ,x)
(Y, Ff (x))
(f,Ff |x )
6 (f,x)
(X, x)
/
(Y, y) O
(idY ,y)
(Y, y)
Figure 3.10: Concrete isomorphisms between (F, C, Set), X oF C,
R C
¯ F
have a product category (C × Set) (for simplicity we will assume these are constructs but the results hold for any underlying category X not necessarily Set). The basestructured categories are each concrete subcategories of the (C × Set). The concrete R ¯ isomorphism is established by noting that objects of (F, C, Set) are identical to C F and X oF C whereas the morphism (f, Ff |x ) : (X, x) → (Y, y) is in bijection with (f, y) : (X, x) → (Y, y) and (f, x) : (X, x) → (Y, y) and concrete isomorphism is established as shown in Figure 3.10. Thus we have, (F, C, Set) ∼ =
Z
¯∼ F = X oF C
(3.7)
C
Indeed all the three categories are isomorphic and have a first projection onto the category C defined as p : (F, C, D) → C where p(X, F X) := X for all objects (X, F X) ∈ Ob((F, C, D)) and p(f, F f ) := f for all morphisms (f, F f ) ∈ mor((F, C, D)). Using Proposition 19 it easily follows that the first projection funcR ¯ to C are the classic split (op)fibrations. On tors sending (F, C, D), X oF C, C F the other hand from Proposition 62 it easily follows that the first projection functors R ¯ X oF C are the classic split opfibrations. sending (F, C, Set), C F, The terminology base-structured categories reflects the fact that these categories have an abstract isomorphism with the base category C. In base-structured categories the objects of D are trivially categorified which means in essence the structure only coming from the base objects and arrows. It is only when we consider the objects F X (and therefore arrows F f ) as non trivial either as structured sets (which enables us to make use of set-theory along with category theory) or itself as category (which enables us to continue in category theory as classic fibrations with non-trivial vertical arrows). c 2018, Indian Institute of Technology Delhi
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Since this is an obvious specialization of fibred categories it seemed appropriate to us to call the entire family consisting of these three categories as base-structured categories where the base is C; since all arrows are Cartesian the essential structure of these categories is that of the base. We will have more to say on these base-structured categories with their distinct perspective of a functor in applying category theory in fundamental applications such as symmetries and geometries in Chapter 5 and signal representation leading to arrow-theoretic redundancy in Chapter 4.
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Chapter 4
Functorial signal representation
In this chapter we introduce and study functorial signal representation along with various cases of L0 and L2 functors. Underlying the functorial signal representation is the basic concept of a functor. In Chapter 3 we studied base structured categories which characterize a functor using two different perspectives viz. the graph of a functor and action of category on family of objects. These categories were the result of efforts to combine the intuitions of Leyton’s generativity and Grothendieck’s relativity. They are new to basic category theory leading to an elementary concept of trivial categorification as discussed in Section 3.2.1. Further being special cases of fibred categories resulting from a perspective of making only base structure explicit through category theory concealing the vertical structure using identity morphisms combined intuitions of Grothendieck’s relative and Leyton’s generative theory. In this chapter we shall see how these have facilitated the application of category theory in signal representation along with set-theoretic measure theory. We visually motivate the functorial viewpoint of signal using Figures 4.1 and 4.2 and then we summarize the major differences of proposed functorial framework from classic representation in Table 4.1 which stand as prime motivations for a functorial representation of signals discussed in Section 4.1. The fundamental concept of treating a signal as functor is not entirely new and is first explored in the context of topology in [Rob14]. However, there are some subtle differences in this work as discussed in Section 4.1 especially the underlying generative cause. Nevertheless the work in [Rob14] stands as third motivation in addition to primary category-theoretic relative perspective of [Gro71] and generative intuition of [Ley01] towards developing this framework.
4.1 Functorial signal model
4.1
83
Functorial signal model
In this section we first show intuitive Figures 4.1, 4.2 for grasping the new framework more easily. The major differences of proposed functorial framework from classic representation summarized in Table 4.1 are discussed in detail then using Figure 4.3 and Equation 4.1. Note that one can formulate all these differences alternatively using Equation 4.2 instead of Equation 4.1 by substituting F (G1 ) = G1 l, F (G2 ) = G2 l, F (a) = (aM , ao ) = (h, φ) using the fact that Meas2 is same as Meas→ appropriately as discussed in Section 4.1.9. Figure 4.1 provides a visual depiction of proposed functorial signal representation framework, the salient features of which are as follows: 1. Complete signal (a measurable function) f = ... q (f I) q (f K) q (f J) q ... is naturally the coproduct in category Meas→ or functor category Meas2 , where subobjects (f I), (f K), (f J), ... are local real-valued partial functions on disjoint half-open intervals I, J, K, .... 2. The underlying generators(capturing the intuition of generative theory in [Ley01]) of a signal are directly modeled either as functors G1 ,G2 ,G3 ,... or else as objects of base category C. The transfers (or isomorphisms) between the generators are captured via natural transformations (or natural isomorphisms) or via base category arrows a1 , a2 , .... 3. Then functorial representation models the signal either as a functor F : C → D or as a subcategory of the usual functor category Meas2 4. Whenever G1 and G2 are isomorphic, the corresponding subobjects (f I), (f J) also become isomorphic via (h, φ) provided the functor preserves this isomorphism. Then (f J) is naturally viewed as redundant relative to (f I). 5. By considering a field structure or property on the underlying set of R, the objects of Meas→ such as (f I) are also set-theoretically elements of Riesz spaces L0I additionally bringing familiar set-theoretic measure-theory alongside category theory for signal representation. Properties of null-ideal, measure on (I, ΣI ) allows introducing equivalence classes (f I)• ∈ L0 (I, ΣI , N (µI )), (f I)• ∈ L2 (I, ΣI , µI ) through set-theoretic measure-theory. 6. The arrow (h, φ) in certain cases induces T(h,φ−1 ) : L2 (I, ΣI , µI ) → L2 (J, ΣJ , µJ ) and signal space becomes a subcategory in Hilb or Riesz matched to its generative structure as shown in Figure 4.2. c 2018, Indian Institute of Technology Delhi
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R
84
Global space L0R
f (t)
Subspace L0I
Subspace L0K
Subspace L0J
f I = F (G1 )
f K = F (G3 )
f J = F (G2 )
(0, 0)
t R I
J
K
Figure 4.1: Functorial signal representation: Global Signal f is chopped into naturally related local sub-signals ..(f I),(f K),.. thought of lying in a category.
T(h,φ−1 )
L2 (I, ΣI , µI )
L2 (K, ΣK , µK )
(f I)•
0
L2 (J, ΣJ , µJ )
(f K)•
(f J)•
0
0
Figure 4.2: Signal space as a subcategory of Hilb or Riesz
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4.1 Functorial signal model
Contemporary function based model Signal as entity varying in time or space modeled mathematically as a function Simple 1D time signal as measurable ˆ B ) → (R, ΣB ) function f : (R, Σ ˆ B ) → (R, ΣB ) viewed as a special case Signal f : (R, Σ F : C → Meas→ ; C being some discrete category. Space L0 R (µ) canonically isomorphic Q to i∈(I,J,..) L0 i (µi ) via f 7→ (f I, f J, ...) Using properties of =a.e. , measure, |f |2 integrable, ˆ B , N (µ)), f • ∈ L2 (R, Σ ˆ B , µ) field R, f • ∈ L0 (R, Σ Signal space is generically fixed ˆ B , N (µ)) or L2 (R, Σ ˆ B , µ). as L0 (R, Σ ˆ B , µ) in Hilb Signal space is a fixed object L2 (R, Σ or Riesz where L2 : LocMeasure → Riesz When C = 1, time axis thought of as single base ˆ B ) of Meas. corresponding to the object (R, Σ Classic signal f • ∈ L2 (R, ΣLeb , µ) = L2 (µ) f • 7→ [(f I)• , (f J)• , ...] In summary no mathematical modeling of signal source, pure set-theoretic measure theory
85
Proposed functorial model Signal as a functor from a category (modeling generative structure)to a category (modeling observed waveforms) Simple 1D time signal as F C using a functor F : C → Meas→ Signal as a family of objects F (G1 ) = f I, F (G2 ) = f J, ... along with non-trivial arrows F (a1 ), F (a2 ), ... f J = F a(F G1 ) = f I-valued point of f J + ∆J (f I, f J, ...) as family of generalized elements and ∆s Properties of null-ideal,measure on (I, ΣI ), R as field permits (f I)• ∈ L0 (I, ΣI , N (µI )), (f I)• ∈ L2 (I, ΣI , µI ) When arrows (h, φ), (h0 , φ0 ), ... uniquely define operators the resulting subcategory in Hilb or Riesz is signal matched. When C ⊆ LocMeasure is generative category, then generative signal matched space is directly modeled as L2 (C) Multiple (I, ΣI ),(J, ΣJ ),... bases and change of base in opposite Measop leading to celebrated relative viewpoint. When C ⊆ LocMeasure, signal f • = [(f I)• , (f J)• , ...] (f J)• represented as ∆J + L2 (φ−1 )(f I)• In summary mathematical modeling of source as category combining both category and set-theoretic measure theory
Table 4.1: A summary of major differences between conventional and functorial signal model
In summary, the category-theoretic perspective views signal f = ... q (f I) q (f K) q (f J) q ... as the coproduct object in category Meas→ where the subobjects are related by special arrows (h, φ) : (f I) → (f J). The generative perspective views objects (f I) = F (G1 ) as having correspondence to natural generators G1 such as melodies/physical objects/linguistic words related by arrow a : G1 → G2 modeled using a functor F : C → Meas→ . Finally the Set-theoretic measuretheory perspective utilizes field property on the underlying set of measurable space (R, ΣB ), treating (f I) also as an element of Riesz space L0I . Moreover the classic signal representation techniques do not take into account any special mathematical modeling of signal source and utilize pure set-theoretic measure theory and functional analysis. In the proposed functorial model the mathematical modeling of source as a category facilitates combining both category-theory (especially for relative viewpoint) and set-theoretic measure theory (for computational viewpoint) simultaneously.
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R amplitude f I = F G1
f J = F G2
(0, 0)
time R I
J
Figure 4.3: In complete time signal f (global on its entire duration), the restrictions f I and f J (or local sub-signals) on half-open intervals I, J are generated by objects G1 , G2 of base category isomorphic to each other through arrow a. a
G1
/
F
G2 − →
(R, Σ ) O B
h
/
(R, Σ ) O B F (G2 )∼ =F (a)[F (G1 )]
f I=F (G1 )
(I, ΣI )
(4.1)
φ
/ (J, ΣJ )
G1
$
2
a
Meas :
G1 M = O (R, ΣB ) o G1 l=f I
G2
4.1.1
G1 O = (I, ΣI ) o
aM =h
/
Sc2 = (R, ΣB ) O
(4.2)
G2 l∼ =(h,φ)f I aO =φ
/
Sc1 = (J, ΣJ )
Underlying intuition
The classic model represents a signal as some real R or complex C-valued function1 of time or space denoted as f : R → R. Depending on the mathematical structures such as measure, topology, group on the domain R, tools from measure theory, functional or harmonic analysis are utilized in analyzing the signal. The proposed model represents same signal appropriately as a functor F : C → D from a base category to a codomain category additionally bringing the powerful relative perspective and other tools from category theory. The base category mathematically models the natural generative 1
function f : D → R is a mapping from one set of values the domain D, to another set the codomain R, assigning each member x ∈ D, the value f (x) ∈ R.
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structure [Ley01] of a signal. Its objects correspond to true generators while the arrows capture relationship between them. The codomain category is at times arrow category of another category. This is often the case since signals classically considered as functions are most appropriately modeled as arrows of a category. The study of functor as a structure-preserving-morphism and trivial categorification was done in Chapter 3. The base category interpreted as category of generators and transfers serves to capture intuition of Leyton’s generative theory as in [Ley86a], [Ley01] and will often be a groupoid. Classical function based model Signal as real (or complex) valued entity varying in time or space domains modeled mathematically as a real (or complex) valued function of time or space.
Proposed functorial model Signal as a functor from a category (modeling generative structure)to a category (modeling observed waveforms)
Table 4.2: Difference in underlying intuitions between conventional and proposed signal representation model
4.1.2
Case of basic measurable structure
Considering a measurable structure on the domain R while taking the signal to be realˆ B) → valued the classical model is precisely a Lebesgue2 -measurable function f : (R, Σ (R, ΣB ). The proposed model represents same signal as a functor F : C → Meas→ 3 denoted simply as the image subcategory F C. The work in [Rob14] is the study of signal sheaf F : FaceX → Vect using face category of simplical complex X based on presheaf model of a signal F : OpnX → Vect. These are based on topological structure of signal domain appropriately modeled as base category. In our proposed model F : C → D, the base category C is expected to capture the natural generative structure of signal to be represented; the intuition of which is based on work 2ˆ
ΣB is σ-algebra of Lebesgue measurable sets, the completion of ΣB the σ-algebra of Borel subsets of R. 3 Meas is the category of measurable spaces (X, ΣX ) and measurable functions f : (X, ΣX ) → (Y, ΣY ). Meas→ is the arrow category of Meas with measurable functions f : (X, ΣX ) → (Y, ΣY ) as objects and pairs of measurable functions (h, φ) : f → g as arrows such that h ◦ f = g ◦ φ.
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in [Ley86a], [Ley01]. The major difference in our approach is that the mathematical structure to be considered on time or space domain is a groupoid determined by generative category C specific to the signal and depending on choice of D can vary over multiple structures such as measure (which will be explored here choosing D = Meas→ ), topology and graph which are present in specific classes of signals. Considering a signal generally as F C, leads us towards general arrow-based redundancy, Grothendieck’s relative viewpoint in signal analysis and understanding how compression basically occurs. Classical function based model simple 1D time signal as measurable ˆ B ) → (R, ΣB ) function f : (R, Σ
Proposed functorial model Simple 1D time signal as F C using a functor F : C → Meas→
Table 4.3: Difference in mathematical expressions using most basic measurable structure
4.1.3
Trivial arrows versus non-trivial arrows
ˆ B ) → (R, ΣB ) can be considered as a special case of proThe classic model f : (R, Σ posed model F : 14 → Meas→ (or F : C → Meas→ with C as a discrete category) ˆ B ) → (R, ΣB ) along with idenwhere F (1) is a category with object F (1) = f : (R, Σ tity F (id1 ) = idf (or discrete image subcategory consisting of objects f I, f J, ...). Thus the classic model from the viewpoint of proposed model is essentially a single object f of Meas→ with trivial identity or a discrete subcategory of Meas→ . The proposed model is then viewed as generalization to arbitrary base category. If we denote objects of C as G1 , G2 , ... and the arrows as a, a0 , a00 , ..., then signal is a collection or family of objects F (G1 ) = f I , F (G2 ) = f J, ... and the arrows F (a) = (h, φ), F (b) = (h0 , φ0 ), .... The multiple non-trivial arrows across objects in proposed model truly bring the relative perspective of whole category theory in signal analysis. In arrow category, (h, φ) : f → g where h ◦ f = g ◦ φ; the pair (h, φ) is 4
1 is the trivial category with a single (trivial) object 1 and just identity (trivial) arrow id1 .
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not unique but general and in particular specifying f → g makes it unique in that context. Hence we can perfectly have (h, φ) : f 0 → g 0 where h ◦ f 0 = g 0 ◦ φ and therefore it is always understood with particular domain and codomain. Classical function based model ˆ B ) → (R, ΣB ) viewed as a special case Signal f : (R, Σ F : C → Meas→ ; C being some discrete category.
Proposed functorial model Signal as a family of objects F (G1 ) = f I, F (G2 ) = f J, ... along with non-trivial arrows F (a1 ), F (a2 ), ...
Table 4.4: Difference of trivial arrows versus additional non-trivial arrows
4.1.4
Collection of independent elements versus arrow based generalized elements and differentials.
Now referring Figure 4.1, in set-theoretic measure theory the signal f ∈ L0 R is Q simply an element of a global space L0 R canonically isomorphic to i∈(I,J,..) L0 i via f 7→ (f I, f J, ...). Thus the collection (f I, f J, ...) in classic case has local components such as f I which are always linearly independent. This independence in classic model does not authentically reflect the natural generative relationship between these components which make them interdependent typical to sources with memory from information theory viewpoint. The proposed model precisely utilizes the arrows and fundamental concept of generalized elements to represent the components differentially. Referring Equation 4.3, F (G1 ) = f I, F (G2 ) = f J and F (a) = (h, φ) this implies f J is related to f I via arrow F (a) therefore we express f J = f I-valued point of f J + ∆J where ∆J is differential error in actual measured signal. Thus signal becomes a collection (f I, F (a)f I + ∆J , ...) where generalized elements5 and ∆s together faithfully model the interdependencies between these components. Differential ∆J = F (G2 ) − F (a)F (G1 ) in Riesz space L0 J roughly indicates the linear deviation of G2 from G1 and is relatively small when arrow a is (total) isomorphism. Note that vector addition in F (a)F (G1 ) + ∆J is not a coproduct operation in Meas→ 5
A generalized element of N (also called M-valued point of N) is just an arrow a : M → N in a category. Hence an element of a set S in set-theory is simply the arrow e : T → S in the category Set where T is a terminal object.
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but an operation in L0 J putting certain limitations on faithfulness of functor F which we discuss in Section 4.1.10. Note that f J : (J, ΣJ ) → (R, ΣB ) is a measurable function therefore an object of Meas→ . But it is also a trivial category and therefore by considering additional property of R being a field, it is can be point-wise added to and multiplied by any other measurable function gJ : (J, ΣJ ) → (R, ΣB ). In other words, f J is simultaneously an element of Riesz space L0 J and an object of Meas→ which recognizes only measurable structure on R. This is a novel concept of using set-theory alongside category-theory simultaneously for signal representation application.
G1
a
/
G2
F
− →
(R, ΣB ) O
(4.3)
∆J
(R, ΣB )
/O
h
O
f I=F (G1 )
(I, ΣI )
Classical function based model Space L0 R (µ) canonically isomorphic Q to i∈(I,J,..) L0 i (µi ) via f 7→ (f I, f J, ...)
F (a)[F (G1 )] φ
/
(J, ΣJ )
Proposed functorial model f J = F a(F G1 ) = f I-valued point of f J + ∆J (f I, f J, ...) as family of generalized elements and ∆s
Table 4.5: Difference of independent elements versus arrow based generalized elements and differentials.
4.1.5
Signal space as a signal matched category
Using additional properties of equivalence relation =a.e. , Lebesgue measure (µ) and |f |2 being integrable, the classic signal f is usually not differentiated from almost ˆ B , N (µ)) and using equal everywhere measurable functions on R. Thus f • ∈ L0 (R, Σ ˆ B , µ)6 . Thus classic signal measure along with property |f |2 is integrable, f • ∈ L2 (R, Σ Lp (X, ΣX , µ) = Lp = Lp (µ), p ∈ (1, ∞) is set of functions {f • : f ∈ Lp } ⊆ L0 = L0 (µ) in L0 (µ) under =a.e. 6
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f • is a vector in global linear spaces L0 , L2 . Similarly using properties of equivalence relation =a.e. and subspace measure on (I, ΣI ) we have (f I)• ∈ L0 (I, ΣI , N (µI )), (f I)• ∈ L2 (I, ΣI , µI ) while in certain cases (see Propositions 81, 86, 94 for such cases) arrows (h, φ), (h0 , φ0 ), ... uniquely define linear operators (f I)• 7→ (h(f I)φ−1 )• from L2 (I, ΣI , µI ) to L2 (J, ΣJ , µJ ). Then in proposed model signal is thought of lying inside a signal matched category whose objects are L0 (I, ΣI , N (µI )), L0 (J, ΣJ , N (µJ )), ... or L2 (I, ΣI , µI ), L2 (J, ΣJ , µJ ), ... and non-trivial morphisms are operators defined by (h, φ), (h0 , φ0 ). Set-theoretically the signal is a family of elements (vectors) comprising one element from each object of the signal matched category. Further these elements are related by arrows which are nothing but restrictions of the operators within category to individual elements leading to representation with generalized elements and differentials. Classical function based model Using =a.e. , measure, |f |2 integrable ˆ B , N (µ)), f • ∈ L2 (R, Σ ˆ B , µ) f • ∈ L0 (R, Σ Signal space is generically fixed ˆ B , N (µ)) or L2 (R, Σ ˆ B , µ). as L0 (R, Σ
Proposed functorial model Additional properties of null-ideal,measure on (I, ΣI ) permits (f I)• ∈ L0 (I, ΣI , N (µI )), (f I)• ∈ L2 (I, ΣI , µI ) When arrows (h, φ), (h0 , φ0 ), ... uniquely define operators the resulting subcategory in Hilb or Riesz is signal matched.
Table 4.6: Difference of generic function space versus subcategory in Hilb or Riesz.
4.1.6
Invoking functorial L2 construction for special class of inverse-measure preserving maps on measure spaces
In special class of signals where the relationships between local sub-signals are completely captured by the inverse-measure-preserving maps denoted by φ, φ0 , ..., the functorial structure of L2 construction, L2 : LocMeasure7 → Riesz8 or L2 : LocMeasure → Hilb9 can be invoked leading to signal matched category as L2 (C) where C ⊆ LocMeasure. This captures generative structure where generators are related by 7
LocMeasure is the category of localizable measure spaces (X, ΣX , µ) and f • a.e. equivalence classes of inverse-measure-preserving maps f : X → Y 8 Riesz is the category of Riesz spaces and Riesz homomorphisms. 9 Hilb is the category of Hilbert spaces and continuous(bounded) linear maps
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common translations of signal domain. Ofcourse, the classic signal space is fixed ˆ B , µ), a single object in Riesz or Hilb. See real-world examples global space L2 (R, Σ in Section 4.2.2. Classical function based model ˆ B , µ) Signal space fixed L2 (R, Σ 2 where L : LocMeasure → Riesz
Proposed functorial model When C ⊆ LocMeasure for affine transformations of time,then signal matched subcategory is L2 (C)
Table 4.7: Difference of generic function space versus subcategory in Hilb or Riesz.
4.1.7
Change of base and Grothendieck’s relative point of view
By using the opposite category of (complete) Boolean Algebras, it is possible to think of local sub-signals as determined by pullbacks which are unique upto isomorphism, in cases where Boolean homomorphism φ−1 determined by φ on measurable spaces is an isomorphism. In such cases the pullback object ΣB ×ΣI ΣJ is isomorphic to ΣB via some isomorphism h−1 as opposite arrow of some Borel measurable function h : (R, ΣB ) → (R, ΣB ). Thus the opposite arrow g −1 of observed measurable function g is pulled back map of φ−1 ◦ g −1 determined upto isomorphism using g −1 = h−1 ◦ ((φ−1 ◦ g −1 )∗). The change of base from (ΣI , 4, ∩) to (ΣJ , 4, ∩) along φ−1 leads to a relative perspective namely that (ΣB , 4, ∩) fibred on (ΣI , 4, ∩) via φ−1 ◦g −1 produces an object ΣB ×ΣI ΣJ over (ΣJ , 4, ∩) which is isomorphic to (ΣB , 4, ∩) fibered on (ΣJ , 4, ∩) via g −1 .
(R, ΣB ) O
id gφ
gφ
(I, ΣI )
φ
/
(R, ΣB ) 9
O
g
/ (J, ΣJ )
(ΣB , 4, ∩) o φ−1 ◦g −1
w
(ΣI , 4, ∩) o
id
(ΣB , 4, ∩)
φ−1 ◦g −1 φ−1
g −1 (E7→g −1 [E])
(ΣJ , 4, ∩)
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ΣB e
(4.5) id h−1
g −1
%
−1 )∗
(φ ΣB ×ΣI ΣJ o
!
/
ΣB
(φ−1 ◦g −1 )∗
ΣJ o
Classical function based model When C = 1, time axis thought of as single base ˆ B ) of Meas. corresponding to the object (R, Σ
'
φ−1
/
φ−1 ◦g −1
ΣI
Proposed functorial model Multiple (I, ΣI ),(J, ΣJ ),... bases and change of base [Jac99] in opposite Measop leading to celebrated relative viewpoint.
Table 4.8: Difference of fixed base versus multiple bases leading to application of Grothendieck’s relative point of view in proposed model.
4.1.8
Arrow-theoretic understanding of redundancy and compression
ˆ B ,µ) In classic signal representation since the signal space is a fixed object such as L2 (R, Σ the compression is mathematically explained via choosing a certain basis (or frame) in this space resulting into sparse representation. However the functorial viewpoint gives the freedom of choosing spaces and their transformations respecting or matched with generative structure. The model as a category naturally leads to interpreting redundancy through arrows across objects or sub-signals. As an illustration, referring Figure 4.1 we consider the common redundancies purely due to inverse-preserving maps between local intervals. The general redundancy between f I and f J is captured by F (a) and f J is represented as F (a)f I + ∆J . In translational redundancy this is solely determined by φ : (I, ΣI ) → (J, ΣJ ) which is also additionally measure-preserving by considering additional property of null-ideals and measures. Thus φ : (I, ΣI , µI ) → (J, ΣJ , µJ ) and using signal matched category as L2 (C) where C ⊆ LocMeasure as described in Section 4.4.2; we have functorial represen-
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tation as,
∆J + L2 (φ−1 )(f I)• , ...] f • = [(f I)• , |{z} | {z } | {z } classic relative error generative relative term
(4.6)
where, • (f I)• : Local signal representation using some basis in L2 (I, ΣI , µI ) • (f J)• : Local signal representation using some basis in L2 (J, ΣJ , µJ ) • L2 (φ−1 )(f I)• : Transformed/Transfered local signal from L2 (I,ΣI , µI ) to L2 (J, ΣJ , µJ ) • ∆J = (f J)• − L2 (φ−1 )(f I)• : Differential or error between transformed and observed local signal in L2 (J). ˆ B , µ) is the direct sum of h(i, Σi , µi )ii∈(I,J,...) a family of meaBy noting that (R, Σ 0 ˆ B , N (µ)) and Q sure spaces, the canonical isomorphism between L0 (R, Σ i∈(I,J,...) L (µi ) Q 2 ˆ B , µ) and the subspace {u : u ∈ induces an isomorphism between L2 (R, Σ i∈(I,J,...) L (µi ), P Q 2 1/2 < ∞} of i∈(I,J,...) L2 (µi ). Now the classical signal f • ∈ kuk = i∈I ku(i)k2 ) Q L2 (µ) belongs to a global space L2 (µ) canonically isomorphic to subspace of i∈(I,J,..) L2 (µi ) via f • 7→ [(f I)• , (f J)• , ...]. In classical representation the components (f I)• , (f J)• , . . . are always linearly independent and since all these belong to a global space the entire signal is expressed as single element (f I)• + (f J)• + ... where vector addiˆ B , µ) and components such as (f I)• are automatically tion is in global space L2 (R, Σ ˆ B , µ). However thought of having zero value outside restricted interval on whole (R, Σ the same components in proposed representation belong to different sub-spaces L2 (µi ) which are treated as separate objects, related by special arrows in category Riesz or Hilb. These arrows make the components dependent or related while differentials together with arrows are the real innovations which are often encoded. When the arrows L2 (φ−1 ) are compactly represented using φ−1 then L2 being fixed construction overall gain in encoding results since one encodes differential and φ−1 to generate ∆J +L2 (φ−1 )(f I)• instead of encoding (f J)• independently as in classic case. In various lossless compression standards such as PNG or DPCM in lossless JPEG the map φ is fixed specified by the filter type. This provides a category-theoretic mathematical c 2018, Indian Institute of Technology Delhi
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explanation for achieving enhanced compression in differential encoding such as PNG or DPCM in lossless JPEG as compared to standard lossy JPEG using wavelets or discrete cosine transforms especially in certain class of images such as line drawings, text, and iconic graphics involving frequent isomorphisms between neighboring pixels as illustrated using examples in Section 4.2.2 Classical function based model Classic signal f • ∈ L2 (R, ΣLeb , µ) = L2 (µ) f • 7→ [(f I)• , (f J)• , ...]
Proposed functorial model When C ⊆ LocMeasure, signal f • = [(f I)• , (f J)• , ...] (f J)• represented as ∆J + L2 (φ−1 )(f I)•
Table 4.9: Arrow-theoretic Redundancy, compression and relative information content of Signal.
4.1.9
Equivalent formulation using Meas2
So far we have discussed the functorial framework using a basic functor however in this section we formulate an equivalent viewpoint using the higher categorical concept of functor category BC referring [Mac98]. This viewpoint has the advantage of not having to deal with abstract base category as this is often not given in the application but practically we have only concrete observed signals. Since generators (capturing the intuition of generative theory in [Ley01]) of a signal generate concrete measurable waveforms therefore, rather than working with abstract category C we directly model them as functors G1 , G2 , G3 , ... : 210 → Meas. The transfers (or isomorphisms) between the generators then automatically become natural transformations (or natural isomorphisms) a1 , a2 , ... forming some signal matched subcategory (or groupoid) of the usual functor category [Mac98] namely Meas2 . Thus corresponding to objects (f I), (f K), (f J), ... which are local real-valued partial functions on disjoint halfopen intervals I, J, K, .... we associate generators as functors while transfers between generators take the form of natural transformations. From the signal representation point of view both these ways are equivalent and a reader can more or less substitute 10
2 is the category with two objects O, M with identity morphisms and one non-trivial morphism l : O → M.
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one for other using the fact that functor category Meas2 is same as Meas→ , using Equation 4.2 along with arguments presented in Section 4.1.1 to Section 4.1.8. Whenever the category C is not known apriori or abstract, in such cases the generators and their relationships can be concretely modeled as functors G1 : 2 → D and natural transformations a : G1 → G2 .
(β·α)c2 R
2
α / # Meas = β
S T
t
Rc2 = (R, ΣB ) o O
αc2 =h1
/ Sc2
Rf =f1
= (R, ΣB ) o O
βc2 =h2
Sf =f2
/
Rc1 = (I, ΣI ) o j
αc1 =φ1
/
Sc1 = (J, ΣJ ) o
*
T c2 = (R, ΣB ) O T f =f3
βc1 =φ2
/
T c1 = (K, ΣK ) 4
(β·α)c1
(4.7) The generators (capturing the intuition of generative theory in [Ley01]) of a signal generate measurable waveforms therefore can be modeled as functors R, S, T, .... The transfers (or isomorphisms) between the generators then automatically become natural transformations (or natural isomorphisms) α, β, ... forming some signal matched subcategory of the usual functor category Meas2 in [Mac98].
4.1.10
Dealing partial isomorphisms, practical observation errors and general limitations.
In this work, the category for natural generative structure (or equivalently the subcategory of functor category with functors 2 → Meas) is taken as a groupoid since isomorphism most appropriately models the transfer of previously occurring object in the generative theory of of [Ley01]. Moreover the codomain of an isomorphism can be completely determined from its domain and serves to model redundancy of structures and signals in particular. Thus from a given observed signal thought of as a functor (or equivalently as a coproduct of measurable functions on disjoint half-open
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intervals), the goal is to reversely determine its generative groupoid structure. This functor model is generally faithful for groupoids whose objects generate waveforms with disjoint supports. In such cases from the image subcategory we can infer the isomorphic domain category. However when the objects in generative category generate waveforms with overlapping supports then faithfulness breaks down as from the category perspective superposition operation is not a coproduct operation. This introduces limitations on the generative groupoid which can be inferred from observed signal as described below: 1. Non-faithfulness of functor: In case of generators producing waveforms or measurable functions whose domain intervals are not naturally disjoint, the functor becomes non-faithful or in other words some structure is lost and therefore all isomorphisms are not preserved. This occurs since observed signal is a superposition of these waveforms through additive field property on R which is not a coproduct operation in Meas→ or a coproduct in functor category Meas2 . Common examples include combination of melodies in music signal or two different speakers uttering words simultaneously in speech signal. 2. Structures recognized in a category: The constraints imposed by choice of codomain category which determine the type of isomorphisms that can be modeled. In the case of Meas→ using isomorphisms of type (h, φ) only measurable structures on the generators can be modeled through isomorphisms which includes translations, scaling, amplitude changes but not any other structure such as topology of images etc. 3. Practical limitations of generating systems: Due to practical limitations of equipments it is quite possible that perfectly isomorphic generators may produce slightly dissimilar waveforms. While if there is a partial isomorphism between natural generators of the signal or in other words the generative category is a partial groupoids then certainly we have correspondingly dissimilar waveforms. These limitations of the functorial model can be dealt with in different ways as we discuss now. Note since direct sum is the coproduct in Meas we have for disjoint half-open intervals I and J the coproduct of two L L measurable functions f1 and f2 L L f1 f2 f1 f2 I− →R J− →R ∼ J −−−−→ R R . = I In the first limitation, if f1 , f2 are two measurable functions (corresponding to two generators) the domains of which intersect each other then using we have three c 2018, Indian Institute of Technology Delhi
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disjoint measurable spaces domf1 \ domf2 , domf2 \ domf1 , domf1 ∩ domf2 . Correspondingly we have three functions f1 (domf1 \ domf2 ), f2 (domf2 \ domf1 ), (f1 + L f2 )(domf1 ∩domf2 ) whose coproduct (direct sum) f1 (domf1 \domf2 ) (f2 (domf2 \ L domf1 ) (f1 + f2 )(domf1 ∩ domf2 ) is the observed signal to be represented. Thus the two generators get treated as being three corresponding to the three measurable functions on disjoint intervals and can be recovered or filtered only in special cases as discussed in third limitation. Extending to overlapping of finite n generators we have f1 , f2 ,...,fn as measurable functions of which we can similarly form coproduct of disjoint measurable functions. The second limitation is dealt with by varying the codomain category D to categories of sheaves, graphs, topological measurable spaces etc. This can be corresponding to structures such as topology, graphs depending on specific class of signals we need to represent where a particular structure is prevalent. The third limitation is on account of the arrows which recognize only measurable structure on R but don’t respect the field structure thereby restricting faithful functorial modeling of partial groupoid as the generative category. Workaround for this limitation is using set-theoretic measure theory and considering the usual linear structure of the signal spaces such as L0 X , L0 X or L2 X which take into account the field structure on R. Referring Equation 4.8, if the isomorphism is partial which means G2 is not totally isomorphic then differential ∆J = F (G2 ) − F (a)F (G1 ) in Riesz space L0 J serves to indicate linear deviation of G2 from G1 . In case of a being total isomorphism, ∆J also directly indicates the difference between theoretical local signal F (a)F (G1 ) from practical observed signal F (G2 ). Only when the generating and sensing equipments are ideal we will have theoretically the signal F (a)F (G1 ) (corresponding to object (a)(G1 )) equal to the observed signal F (G2 ) (corresponding
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to object G2 ).
G1
/
a
F
G2 − →
(R, Σ ) O B
(4.8)
∆J
(R, Σ ) O B
/O
h
f I=F (G1 )
F (a)[F (G1 )]
(I, ΣI )
/ (J, ΣJ )
φ
When ∆J is relatively very small as compared to F (a)F (G1 ) it can be attributed to practical limitations of equipments and we can infer total isomorphism between G1 and G2 . In case of partial isomorphism between G1 and G2 , ∆J is quite large. In this case we can continue to theoretically treat arrow a in the base as total isomorphism and then operate on ∆J as new local signal corresponding to subobject of G2 apart from remaining subobject (a)(G1 ) isomorphic to G1 as illustrated in the next example.
4.1.11
A prototypical example beneath the design of Portable network graphics. f [n] 4
2
1
2
3
4
5 n
Figure 4.4: Sequence f [n] = [1, 2, 3, 4, 5, ...]: A prototype differential coding example underlying PNG. c 2018, Indian Institute of Technology Delhi
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As shown in Figure 4.4, consider a signal as sequence of real values 1, 2, 3, .... terminating at some finite n. This well-known sequence is encoded using differential coding by subtracting from each value the previous occurring value to obtain highly compressible sequence 1, 1, 1, .... This example can be recast in a functorial way making use of the partial groupoid structure in base category. The example is specifically used to illustrate the limitation where functor is non-faithful and therefore we invoke set-theoretic measure theory to provide a work-around. (R, Σ ) O B ∆K =f [3]−(id,φ2 )f [2]
(R, ΣB )
id(R,ΣB )
O
/O
∆J =f [2]−(id,φ1 )f [1]
(R, ΣB )
id(R,ΣB )
/O
O
(id,φ1 )f [1]
f [1]
({1}, {∅, {1}})
G1
(id,φ2 )f [2]
φ1
a1
/
({2}, {∅, {2}}) / G0
2
q G002
/
φ2
a2
/
({3}, {∅, {3}}) G03 q G003 q G000 3
Figure 4.5: The functorial signal representation model for f [n] for prototype (Identities and composite arrows are not shown). We utilize a subsequence of just three samples f [1], f [2], f [3] to illustrate the limitations and workarounds as discussed in Section 4.1.10. Consider a subgroupoid within the category Set with six singletons as objects G1 , G02 , G002 , G03 , G003 , G000 3 and isomorphisms a1 , a2 , a02 , A1 , A2 . Note that we have not considered identities and compositions in this groupoid explicitly as they are not relevant to limitations we are discussing; however they are present implicitly. Now define a functor F which takes these objects and arrows to measurable functions as illustrated in Table 4.10. Since the generators generate measurable functions whose supports are not completely disjoint the observed signal is sequence f [1] = g1 , f [2] = g2 + ∆J , f [3] = g3 + g30 + ∆K where the field addition is not a coproduct operation in Meas→ . This forces us to reduce the base category to discrete groupoid with objects G1 , G02 q G002 , G03 q G003 q G000 3 corresponding to measurable functions f [1], f [2], f [3] if we want functor to be c 2018, Indian Institute of Technology Delhi
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isomorphism-preserving. Fortunately using Equation 4.8 as discussed in workaround for limitations we can bring the set-theory once again to undo the field addition using inverses or subtraction operation. Utilizing the fact that G02 q G002 is partially isomorphic to G1 we can represent this isomorphic subobject as F (a1 )F (G1 ) and recover the remaining subobject as ∆J = F (G02 q G002 ) − F (a1 )F (G1 ) as shown in Figure 4.5.
G000
A2
~ a 1 /
G1 o
G02 o
a2
B O
GO 003 /
(4.9)
∆K
~ a0 2 00 o / G > O2
A1
(R, ΣB )
> O3
(K, ΣK ) B
(R, ΣB ) o
G03
/
id
(R, ΣB )
B O
∆J
(J, ΣJ ) o B
(R, ΣB ) o
id
O
g1
(R, ΣB ) o
/
(K, ΣK )
φ2
/
id
O
(R, ΣB ) O
g2
(I, ΣI ) o
/
O
g30
φ1
/ (J, ΣJ ) o
g3
φ2
/
(K, ΣK )
Thus in case of generators such as G1 and G2 = G02 qG002 being partially isomorphic either they are treated as having no interconnected isomorphism or as being connected through a (partial) isomorphism such as a : G1 → G2 as shown in Figure 4.6. In that case we represent the measurable function F (G2 ) = F (a1 )F (G1 ) + ∆J interpreted as generalized element with differential. Such a model using partial isomorphism a1 can be utilized especially when the sequence is expected to be highly structured in the sense that differentials themselves are also related to each other as in this prototype example. Since the relationships are completely determined by fixed maps φ1 , φ2 , ... which c 2018, Indian Institute of Technology Delhi
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Functorial framework parameters Base Category C Objects Base Category C Arrows
Image subcategory F (C) Objects
Image subcategory F (C) Arrows
102
Values for differential coding example G1 , G02 , G002 , G03 , G003 , G000 3 model ∆ , ∆K since example is highly structured. G002 , G000 J 3 a1 , a2 , a02 , A1 , A2 A1 , A2 model ∆J , ∆K inter-relationship in this example. F (G1 ) = g1 : (I, ΣI ) → (R, ΣB ) where (I, ΣI ) = ({1}, {∅, {1}}) F (G02 ) = g2 : (J, ΣJ ) → (R, ΣB ) where (J, ΣJ ) = ({2}, {∅, {2}}) F (G03 ) = g3 : (K, ΣK ) → (R, ΣB ) where (K, ΣK ) = ({3}, {∅, {3}}) g1 , g2 , g3 are all measurable functions each with value 1 ∈ R F (G002 ) = ∆J : (J, ΣJ ) → (R, ΣB ) where (J, ΣJ ) = ({2}, {∅, {2}}) F (G003 ) = g30 : (K, ΣK ) → (R, ΣB ) where (K, ΣK ) = ({3}, {∅, {3}}) F (G000 3 ) = ∆K : (K, ΣK ) → (R, ΣB ) where (K, ΣK ) = ({3}, {∅, {3}}) ∆J , g30 , ∆K are all measurable functions each with value 1 ∈ R . F (a1 ) = (id, φ1 ) : g1 → g2 such that id ◦ g1 = g2 ◦ φ1 or g2 = g1 ◦ φ−1 1 F (a2 ) = (id, φ2 ) : g2 → g3 such that id ◦ g2 = g3 ◦ φ2 or g3 = g2 ◦ φ−1 2 F (a02 ) = (id, φ2 ) : ∆J → g30 where id ◦ ∆J = g30 ◦ φ2 or g30 = ∆J ◦ φ−1 2 φ1 , φ2 are all measurable functions each being isomorphism. F (A1 ) = (id, φ1 ) : g1 → ∆J such that id ◦ g1 = ∆J ◦ φ1 or ∆J = g1 ◦ φ−1 1 F (A2 ) = (id, φ2 ) : ∆J → ∆K such that id ◦ ∆J = ∆K ◦ φ2 or ∆K = ∆J ◦ φ−1 2 ∆J , ∆K only in this example are identical with g2 g30 due to special structure.
Table 4.10: The differential coding prototype example underlying the image enconding standards such as PNG. (R, Σ ) O B ∆K =f [3]−(id,φ2 )f [2]
(R, Σ ) O B
id(R,ΣB )
/O
∆J =f [2]−(id,φ1 )f [1]
(R, Σ ) O B
id(R,ΣB )
f [1]
/O (id,φ1 )f [1]
({1}, {∅, {1}})
G1
(id,φ2 )f [2]
φ1
a1
/
({2}, {∅, {2}}) /
G2
φ2
a2
/
({3}, {∅, {3}}) /
G3
Figure 4.6: The functorial signal representation model for f [n] for sequence (Identities and composite arrows are not shown). are also inverse-measure-preserving isomorphisms by considering counting measure on all measurable spaces which are finite sets with a singleton. Hence functorial signal
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representation is directly given by Equation 4.6 which becomes,
2 −1 2 −1 Signal = [f [1], ∆J + l2 (φ−1 1 )f [1], ∆K + l (φ2 )[∆J + l (φ1 )f [1]], ...]
(4.10)
since in this case, • (f I)• = f [1] : Local signal using basis eˆI in L2 (I, ΣI , µI ) = l2 ({1}, {∅, {1}}, count) • (f J)• = f [2] : Local signal using basis eˆJ in L2 (J, ΣJ , µJ ) = l2 ({2}, {∅, {2}}, count) • (f K)• = f [3] : Local signal using basis eˆK in L2 (K, ΣK , µK ) = l2 ({3}, {∅, {3}}, count) 2 2 • l2 (φ−1 1 )f [1] : Transformed local signal from L (I, ΣI , µI ) to L (J, ΣJ , µJ )
• ∆J = f [2] − l2 (φ−1 1 )f [1] : Error between transformed and observed local signal 2 in L (J). • ∆K = f [3] − l2 (φ−1 1 )f [2] : Error between transformed and observed local signal 2 in L (K).
The functorial framework mathematically makes precise the underlying intuition behind working of the classic filtering phase of all major image encoding standards utilizing differential coding techniques. Especially if the generative category is such that overall most of the generators are isomorphic to their preceding generators then this type of differential filtering will result into almost sparse sequence the reason for which was illustrated through the prototype example. For practical examples see Section 4.2.
4.2
Redundancy and examples
In this section, we contribute a novel category-theoretic Definition 73 of intra-signal redundancy and some of its special cases especially Definitions 74, 75, 76 using the isomorphism arrow in a category. Next we illustrate taking examples of some real-world c 2018, Indian Institute of Technology Delhi
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R
104
f0
f
0 t I
J
Figure 4.7: A global signal where the local signal f = F (G1 ) = G1 l ∈ L2 (I, ΣI , µ) in half-open interval I is translated to f 0 = F (G2 ) = G2 l ∈ L2 (J, ΣJ , ν) in half-open interval J images how compression occurs. The well-known heuristic yielding better compression of PNG standard in image types such as line drawings, iconic image, text etc. compared to classic JPEG standard at a given SNR can be precisely explained using category-theory. Finally we comment on base-structured category perspective of signals and their spaces leading to possible unification of various existing techniques of representing signals.
G1
a
/
F
G2 − →
(R, Σ ) O B
h
/
(R, Σ ) O B
(4.11)
F (G2 )∼ =F (a)[F (G1 )]
f I=F (G1 )
(I, ΣI )
φ
/ (J, ΣJ )
G1
$
2
a
G2
Meas :
G1 M = (R, ΣB ) o
aM =h
O
G1 l=f I
G1 O = (I, ΣI ) o
/
Sc2 = (R, ΣB ) O
(4.12)
G2 l=f J aO =φ
/ Sc1
= (J, ΣJ )
Using Figure 4.7 and Equations 4.12, 4.11 we give the following definitions:
Definition 73 Consider a complete signal modeled as some subcategory of a functor category Meas2 with functors G1 , G2 , ... and natural transformations a1 , a2 , ..., where 2 is the category with two objects O, M with identity morphisms and one non-trivial c 2018, Indian Institute of Technology Delhi
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morphism l : O → M and Meas is the category of measurable spaces and measurable functions. A local signal G2 l corresponding to the generator G2 : 2 → Meas is defined as redundant relative to a local signal G1 l corresponding to the generator G1 : 2 → Meas iff there exists a isomorphism (h, φ) : G1 l → G2 l between them, resulting from the natural isomorphism a : G1 → G2 between its generators. Alternatively, given a signal modeled as some faithful isomorphism-preserving functor F : C → Meas→ with generators G1 , G2 , ... and transfers a1 , a2 , ... as objects and arrows of C, a local signal F (G2 ) is defined as redundant relative to local signal F (G1 ) iff there exists an isomorphism F (a) : F (G1 ) ↔ F (G2 ) which the image of isomorphism a : G1 → G2 in C.
4.2.1
Special cases of redundancies
The specific cases of isomorphisms (h, φ) lead to three special type of redundancies common in signal theory. Let I = [a, b) be an interval and J = [a + T, b + T ) be its translation on real line by T ∈ R. More precisely this means there exists a map φ : I → J given by φ(i) = i + T, i ∈ I. Now φ is both one-to-one and onto or bijection of sets, hence its inverse is given by φ−1 (j) = j − T, j ∈ J. Moreover by using the result that if set A is measurable then for any T ∈ R, A + T = {a + T ; a ∈ A} is measurable, we conclude that both φ and φ−1 are measurable. Also by using Lebesgue subspace measures µI , µJ on I and J, φ, φ−1 are both inverse-measure-preserving since translation leaves Lebesgue measure invariant leading to following definition:
Definition 74 Consider a complete signal modeled as some subcategory of a functor category Meas2 with functors G1 , G2 , ... and natural transformations a1 , a2 , ..., where 2 is the category with two objects O, M with identity morphisms and one non-trivial morphism l : O → M and Meas is the category of measurable spaces and measurable functions. A local signal G2 l corresponding to the generator G2 : 2 → Meas is defined as translation redundant relative to a local signal G1 l corresponding to the generator G1 : 2 → Meas iff there exists a natural isomorphism (idR , φ) : G1 l → G2 l c 2018, Indian Institute of Technology Delhi
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between them, where the measurable map φ : (I, ΣI ) → (J, ΣJ ) given by φ(i) = i + T, ∀i ∈ I defines the translation of domain of G1 l signal to the domain of G2 l signal. Alternatively, given a signal as faithful functor F : C → Meas→ , a local signal F (G2 ) is defined as translation redundant relative to local signal F (G1 ) iff there exists an isomorphism F (a) = (idR , φ) : F (G1 ) → F (G2 ) between them, where the measurable map φ : (I, ΣI ) → (J, ΣJ ) given by φ(i) = i + T, ∀i ∈ I defines the translation of F (G1 ) signal domain to the domain of F (G2 ).
Let I = [a, b) and J = [Sa, Sb) be its scaling on real line by S ∈ R. More precisely this means there exists a map φ : I → J given by φ(i) = Si, i ∈ I. Again φ is both one-to-one and onto or bijection of sets, hence its inverse is given by φ−1 (j) = (1/S)j, j ∈ J. Moreover by using the result that if set A is measurable then for any S ∈ R, SA = {Sa; a ∈ A} is measurable, we conclude that both φ and φ−1 are measurable. Also by using Lebesgue subspace measure µI on I and if µJ is usual Lebesgue subspace measure on J, then setting νJ = (1/S)µJ we conclude φ, φ−1 are both inverse-measure-preserving leading to following definition:
Definition 75 Consider a complete signal modeled as some subcategory of a functor category Meas2 with functors G1 , G2 , ... and natural transformations a1 , a2 , ..., where 2 is the category with two objects O, M with identity morphisms and one non-trivial morphism l : O → M and Meas is the category of measurable spaces and measurable functions. A local signal G2 l corresponding to the generator G2 : 2 → Meas is defined as affine redundant relative to a local signal G1 l corresponding to the generator G1 : 2 → Meas iff there exists a natural isomorphism (idR , φ) : G1 l → G2 l between them, where the measurable map φ : (I, ΣI ) → (J, ΣJ ) given by φ(i) = Si + T, ∀i ∈ I models the affine transformation of domain of G1 l signal into the domain of G2 l signal. Alternatively, given a signal as faithful functor F : C → Meas→ , a local signal F (G2 ) is defined as affine redundant relative to local signal F (G1 ) iff there exists an isomorphism F (a) = (idR , φ) : F (G1 ) → F (G2 ) between them, where the measurable c 2018, Indian Institute of Technology Delhi
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map φ : (I, ΣI ) → (J, ΣJ ) given by φ(i) = Si + T, ∀i ∈ I models affine transformation of the domain of F (G1 ) signal into the domain of F (G2 ) signal.
Finally additionally using a linear Borel measurable function h : (R, ΣB ) → (R, ΣB ) in addition to measurable function φ as above we can model amplified or attenuated redundant signal.
Definition 76 Consider a complete signal modeled as some subcategory of a functor category Meas2 with functors G1 , G2 , ... and natural transformations a1 , a2 , ..., where 2 is the category with two objects O, M with identity morphisms and one non-trivial morphism l : O → M and Meas is the category of measurable spaces and measurable functions. A local signal G2 l corresponding to the generator G2 : 2 → Meas is defined as affine amplified/attenuated redundant relative to a local signal G1 l corresponding to the generator G1 : 2 → Meas iff there exists a natural isomorphism (h, φ) : G1 l → G2 l between them, where the measurable map φ : (I, ΣI ) → (J, ΣJ ) given by φ(i) = Si + T, ∀i ∈ I and h : (R, ΣB ) → (R, ΣB ) being a linear Borel measurable function both together model the affine transformation of domain of G1 l signal into the domain of G2 l signal along with amplification/attenuation of G1 l into G2 l. Alternatively, given a signal as faithful functor F : C → Meas→ , a local signal F (G2 ) is defined as affine amplified/attenuated redundant relative to local signal F (G1 ) iff there exists an isomorphism F (a) = (h, φ) : F (G1 ) → F (G2 ) between them, where the measurable map φ : (I, ΣI ) → (J, ΣJ ) given by φ(i) = Si + T, ∀i ∈ I and h : (R, ΣB ) → (R, ΣB ) being a linear Borel measurable function both together model the affine transformation of domain of F (G1 ) signal into the domain of F (G2 ) signal along with amplification/attenuation of F (G1 ) into F (G2 ).
4.2.2
Examples: groupoid base structures in signals
In this section, first we demonstrate the inherent natural generative groupoid structures in real-world image and audio signals. Figure 4.8 is a sample iconic image of an c 2018, Indian Institute of Technology Delhi
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Figure 4.8: A simple iconic image of umbrella [id] umbrella while Figure 4.10 is another sample image of a real cameraman. By assigning an underlying generator corresponding to each pixel value one can immediately notice that each image is generated by maximal transfer (isomorphism) of existing generators in the language of [Ley01]. This is often the case with line drawings, iconic image, text etc. Also in some real-world photographic images there is a lot of groupoid structure resulting from maximization of transfer such as in the case of background sky, the grass and coat of cameraman. New images as in Figures 4.9, 4.12, are 3D surface plots while Figure 4.12 is 2D plot of images produced by subtracting from each pixel value the previous occurring neighboring pixel value often termed as differential coding to obtain highly compressible or sparse images as observed. These signals are representations of original signals by using a signal-space as the category L2 (C) as where C ⊆ LocMeasure as discussed in Sections 4.1.5, 4.1.6 and Equation 4.10. Notice that only where isomorphisms of the underlying generators break down or in other words generators not related to each other, non-sparse signal values ∆I , ∆J , ∆K , .... are observed such as near the boundary of umbrella. Thus using category-theory we can explain the known heuristic yielding better compression of PNG standard in image types such as line drawings, iconic image, text etc. compared to classic JPEG standard at a given SNR. However such kind of transfers are also inherently present in other kinds of signals such as music since the underlying melodies are maximally transfered by artists as argued in [Ley01]. We illustrate these groupoid structures in a real-world sample audio BBC countdown compilation signal as shown in Figure 4.13. The three marked rectangles illustrate the translation transfer of particular melody twice in a window of 30 seconds as shown superimposed on each-other in Figure 4.14
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Figure 4.9: Surface 3D plot of filtered umbrella image
Figure 4.10: Sample photographic cameraman image [cam]
4.2.3
A base structured category perspective
By taking into account that every signal of interest has an appropriate generative model through intuition of [Ley01] which we have mathematically modeled as some (abstract) category C, the signal model becomes functorial or precisely as F : C → D. c 2018, Indian Institute of Technology Delhi
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Figure 4.11: 2D plot of filtered cameraman
Figure 4.12: Surface 3D plot of filtered cameraman In Chapter 3 we argued that the construction (F, C, D) offers a distinctly refreshingly new perspective on any ordinary functor F : C → D. It is the mathematical expression which combines the intuitive generative perspective of Leyton’s Theory in Psychology [Ley86a], [Ley01] along with well-known Grothendieck’s relative point of view [nLWd] by treating it as a fibration. The distinct perspective was that while general fibred categories carry both base category as well as fibre category structure c 2018, Indian Institute of Technology Delhi
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Figure 4.13: An audio signal of BBC News countdown with sample translation transfer of local melodies in boxed rectangles [Low]
Figure 4.14: Transfered melodies in sample audio superimposed. through cartesian and vertical arrows, the base structured categories carry only the base structure concealing the vertical structure within the objects since the vertical arrows are just identity arrows. Recalling that every functor being a structure preserving morphism we can model c 2018, Indian Institute of Technology Delhi
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same signal equivalently as the base structured category (F, C, D) where the first component of its objects and morphisms signifies natural generators and transfers in generative theory of [Ley01]. However by considering objects F (G1 ), F (G2 ), ... settheoretically as vectors in linear spaces, it becomes possible to view the signal spaces as concrete categories. We illustrate in the case of signal as a functor F : C → Meas→ or equally as the subcategory F (C) of Meas→ . Now we take both set-theoretic and category-theoretic viewpoints simultaneously by noting that (f I) is both an object of Meas→ and an element of Riesz space L0 I using the underlying field property of R. But L0 I is itself an object of category Riesz thus in spirit of classical signal space if we say global signal f ∈ L0 R then functorial signal F (C) belongs to a signalmatched subcategory S of Riesz. The local signal signals (f I) ∈ L0 I , (f J) ∈ L0 J ,... where L0 I , L0 J form objects of that subcategory while the arrows (h, φ) giving rise to Riesz homomorphisms T between these objects in certain cases (see Propositions 81, 86 and 94 for such cases) form the arrows of that subcategory. Often since in classic signal representations we don’t distinguish between f, g ∈ L0 R where f =a.e. g we consider (f I)• by passing to quotient spaces L0 (µI ) or L2 (µI ) (with additional property that |(f I)|2 is integrable) by defining additional functor F 0 : S → Riesz or F 0 : S → Hilb which sends objects such as L0 I to L0 (µI ) (or L2 (µI )) and homomorphisms g 7→ h−1 gφ : L0J → L0I to g • 7→ (h−1 gφ)• : L0 (µJ ) → L0 (µI ) or g • 7→ (h−1 gφ)• : L2 (µJ ) → L2 (µI ). Since there is no standard arrow category in which (f I)• of L0 (µI ) could be considered as an object, we have used an additional functor F 0 . We term category S or F 0 (S) as signal matched space, reflecting the intuitive fact that such a category is compatible with the generative structure of the specific signal to be represented in contrast to a generically fixed space for all signals. As a simple example of translational redundancy in a one-dimensional (global) time signal as shown in Figure 4.7. The functor F : C → Meas→ and duality is
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illustrated in Equation 4.13.
(R, Σ ) O B
id
/
(R, Σ ) O B f 0 =g
f =gφ
(I, ΣI ) G1
φ
a
/
(J, ΣJ ) /
G2
(ΣB , 4, ∩) o f op =πξ
id
(ΣI , 4, ∩) o G1 o
(ΣB , 4, ∩)
π
aop
(4.13)
f 0op =ξ(E7→g −1 [E])
(ΣJ , 4, ∩) G2
In this case the objects F (G1 ), F (G2 ), ... belong to linear spaces L0 of real-valued measurable functions on I and J respectively. Now since a : G1 → G2 in this case is translation, F (a) = (φ, id) and therefore the linear transformation T in opposite direction is purely determined by φ. The spaces L0 of arrows containing F (G1 ), F (G2 ), ... become the objects of such a signal space while the arrows such as F (a), F (a0 ), ... give rise to the transformations of those spaces in contravariant reverse way. In case of a : G1 → G2 purely determined by φ where φ also has the additional property of being inverse-measure-preserving (which includes the popular cases of translation or scaling between I and J) we can consider measures on spaces (I, ΣI ) (J, ΣJ ) and consider equivalence classes f • , f 0• of arrows under =a.e. . Then one can invoke the functorial nature of L0 or L2 as studied in Section 4.4 and signal matched space becomes the image subcategory of L2 |C : C → BanLatt or L2 |C : C → Hilb if we are interested only in linear and norm structure instead of additional lattice and multiplicative structures. In conclusion both local signals such as f I and local spaces L2 (I, ΣI , µI ) containing f I • have dual structures viz the translation structure coming from base arrow and the linear structure coming from the underlying sets with additional properties in contrast to pure linear structure in classic representation techniques.
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Categories for L0, L2 functors
In this section we study in detail the essential categories required for all possible cases of L0 and L2 functors including the special cases of l0 , l2 and their extension to partial categories. These cases are not novel and already scattered throughout literature of both category theory and measure theory; See [Bar92], [Heu13] for category theory, and [Fre11], [Fre16], [Fre12], [Fre13] for a unified treatise of measure theory. Before referring the treatise of measure theory we studied the paper [Heu13] motivating a reader to pursue the case of continuous L2 from the discrete case of l2 on partial categories. Hence we devoted some time to study all possible connections using both total and partial categories (see Section 4.4.3, Appendix A for details on partial categories). From the perspective of signal spaces some simple Definitions 88, 89 would suffice for immediate purpose. However we attempt to cover almost all the cases and their interrelationships in this section as well as in Section 4.4. Simple cases are required in the context of proposed functorial signal representation model when the measure structure of generative base category is of interest for various reasons as motivated in Section 1.1.
4.3.1
Base or domain categories C, Cop
All set-theoretic Lp constructions are inherently functors, see [Fre12]. We will consider two major categories LocMeas a category of localizable measurable spaces refer [nLWb], [Dmi11] LocMeasure a category of localizable measure spaces and their opposites. These categories are useful in context of L0 and L2 (or Lp , 1 ≤ p ≤ ∞). Next we consider special subcategories countMeas,countMeasure and partial monic derived category Par(LocMeas, M), PInj of the earlier categories as summarized in Table 4.11 to cover almost all the cases of l0 and l2 (or lp , 1 ≤ p ≤ ∞). Initially we attempted generalizing the example of l2 functor on the category PInj as studied in [Heu13] through the concept of restriction and inverse categories using underlying category of localizable measurable spaces with finite products. However c 2018, Indian Institute of Technology Delhi
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later referring to the work [Fre12] and references therein we discovered that the functoriality of function spaces is well-studied in the context of Boolean and measure algebras. This made it easier to apply directly in signal representation the perspectives on a functor developed in Chapter 3 to shed light on fundamental concepts of redundancy and compression. Base category C LocMeas LocMeasure countMeas or countLocMeas countMeasure or countLocMeasure PInj, Par(LocMeas, M)
Opposite category Cop LocMeasop LocMeasureop countMeasop or countLocMeasop countMeasureop or countLocMeasureop PInj, Par(LocMeas, M)
Table 4.11: Various base categories covering all cases of functors L0 , L2 , l0 , l2 .
4.3.2
Objects of C, Cop
In this section we discuss the objects of domain categories as tabulated in Table 4.11. Theorem 78 relates the objects of categories LocMeas, LocMeasop , LocMeasure, LocMeasureop . Definition 77 (321I [Fre12]) For a measure space (X, Σ, µ), (B, µ ¯), as constructed in Theorem 78, is called the measure algebra of (X, Σ, µ). Theorem 78 (321H [Fre12]) Let (X, ΣX , µ) be a measure space, and N be the null ideal of µ. Corresponding to the measurable space (X, ΣX , N ) we have B as the Boolean algebra quotient ΣX /ΣX ∩ N . Then we can associate a functional µ ¯:B→ [0, ∞] defined by setting µ ¯E • = µE for every E ∈ ΣX , and (B, µ ¯) is a measure algebra corresponding to (X, ΣX , µ). The canonical map • E 7→ E : ΣX → B is sequentially order-continuous. c 2018, Indian Institute of Technology Delhi
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Proof : We prove this theorem in following steps: (a) From Proposition 34 and its corollary 2, it follows that B is a Dedekind σ-complete Boolean algebra. By Corollary 3, E 7→ E • is sequentially order-continuous, since ΣX ∩ N is a σ-ideal of ΣX . (b) For E, F ∈ ΣX and E • = F • in B, we have E4F ∈ N , hence using µ(E \ F ) = µ(F \ E) = 0 (refer Corollary 1) we get µF ≤ µE + µ(F \ E) = µE ≤ µF + µ(E \ F ) = µF or µF = µE. Consequently the defined formula does indeed define a function µ ¯ : B → [0, ∞]. (c) Finally we verify the axioms of Definition 33. First for the equivalence class of empty set we have µ ¯0 = µ ¯∅• = µ∅ = 0. Second let han in∈N be a disjoint sequence in B. Now choose for each n ∈ N an En ∈ ΣX such that En• = an . Note that hEn in∈N need not be disjoint although En• are equivalence classes and therefore partitions of P ¯an we form a disjoint sequence hFn in∈N by setting ΣX . In order to evaluate ∞ n=0 µ S S • Fn = En \ i 0. Indeed we have verified that (B, µ ¯) is a measure algebra. Objects of category C (X, ΣX , N ) (localizable measurable space) (X, ΣX , µ) (localizable measure space) (X, PX, Φ) (Φ null-ideal of counting measure) (X, PX, counting) (localizable counting measure space) Objects of PInj or Par(LocMeas, M)
Objects of category Cop B (or complete Boolean algebra ΣX /ΣX ∩ N ) (A, µ ¯) (or measure algebra of (X, ΣX , µ)) B (or atomic complete Boolean algebra ΣX ) ¯ (B, counting) (or measure algebra of counting measure on X) Cop via inverse map on arrows if it exists.
Table 4.12: The objects of various base categories from which appropriate functors L0 , L2 , l0 , l2 are defined.
Table 4.12 summarizes all the objects of base categories. The last case of Par(LocMeas , M) is essential for the extension of L0 and Par(B, M) where B ⊆ LocMeasure c 2018, Indian Institute of Technology Delhi
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consists of objects with σ-finite measures caters to the extension of Lp . Then category PInj is same as Par(countMeas, M) for the extended case of l0 but reduces to special case of Par(B, M) where B consists of objects (X, PX, counting) with countable X (since counting measure on a set X is σ-finite iff X is countable) for the extended case of l2 . These extensions were motivated from [Heu13] and make use of the fact that these categories are inverse restriction categories (or roughly speaking partial groupoids) and therefore the arrows are uniquely reversed to form an opposite category from the objects and arrows of original category. This is possible on account of good categorical properties of LocMeas especially the finite products which are required for deriving the partial categories.
4.3.3
Arrows of C, Cop
Now we discuss the appropriate structure preserving functions or homomorphisms of the base categories. Recall that if X and Y are any sets and ΣY a σ-algebra of subsets of Y ; then for an arbitrary function f : X → Y between sets; {f −1 [F ] : F ∈ ΣY } is always a σ-algebra of subsets of X. Now only those functions f are called measurable when {f −1 [F ] : F ∈ ΣY } ⊆ ΣX . It is well known that Meas is a category with measurable spaces such as (X, ΣX ), (Y, ΣY ) as its objects and these measurable functions f as its morphisms. Now the objects of LocMeas and LocMeasure could be viewed as measurable spaces with some added structure such as null ideal or actual measure or dually as appropriate Boolean algebras. Thus the proper structure preserving maps on these objects are measurable functions which preserve this additional structure such as measurable non-singular (measure zero-reflecting) morphisms and Inverse-measure-preserving maps and dually the corresponding Boolean homomorphisms which are reviewed here. Propositions 79, 80 essentially relate the arrows of categories LocMeas, LocMeasop , LocMeasure, LocMeasureop . Proposition 79 (324B [Fre12]) Let (X, ΣX , µ) and (Y, ΣY , ν) be measure spaces, and correspondingly (B, µ ¯), (A, ν¯) their measure algebras. If f : X → Y is a non-singular measurable function such that f −1 [F ] ∈ ΣX for every F ∈ ΣY and µf −1 [F ] = 0 c 2018, Indian Institute of Technology Delhi
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whenever νF = 0. Then there exists a sequentially order-continuous Boolean homomorphism φf : A → B defined by the setting φf F • = f −1 [F ]• for every F ∈ ΣY .
Proof : Let N (µ) and N (ν) be null ideals of the measures µ and ν respectively. Let I = ΣX ∩ N (µ) and J = ΣY ∩ N (ν), then from Theorem 78 we have B and A as the Boolean algebra quotients ΣX /I and ΣY /J respectively. By definition of non-singular measurable function, {f −1 [N ] : N ∈ N (ν)} ⊆ N (µ) hence f −1 [F ] ∈ I for every F ∈ J . Now consider F1 , F2 ∈ ΣY such that F1• = F2• , then F1 4F2 ∈ J consequently f −1 [F1 ]4f −1 [F2 ] = f −1 [F1 4F2 ] ∈ I and f −1 [F1 ]• = f −1 [F2 ]• . So the setting indeed defines a map φf : A → B. Next we prove it is a Boolean homomorphism. For F1 , F2 ∈ ΣY , we have φf F1• 4φf F2• = f −1 [F1 ]• 4f −1 [F2 ]• = (f −1 [F1 ]4f −1 [F2 ])• = f −1 [F1 4F2 ]• = φf (F1 4F2 )• = φf (F1• 4F2• ). Hence φf (a1 4a2 ) = φf a1 4φf a2 for all a1 , a2 ∈ A. Similarly it is easily verified that, φf (a1 ∩ a2 ) = φf a1 ∩ a2 for all a1 , a2 ∈ A, and finally φf 1A = φf Y • = f −1 [Y ]• = X • = 1B . Finally it remains to prove that φf is sequentially order-continuous. To prove this, let han in∈N be a sequence in A. For each n choose an Fn ∈ ΣY such that Fn• = an , and S let F = n∈N Fn . As the map E 7→ E • : ΣY → A is sequentially order-continuous either from Theorem 33 or from Corollary 3, F • = supn∈N an in A. Hence S φf (supn∈N an ) = φf F • = f −1 [F ]• = ( n∈N f −1 [Fn ])• = supn∈N f −1 [Fn ]• = supn∈N φf Fn• = supn∈N φf an , proving φf is sequentially order-continuous. Proposition 80 (324M [Fre12]) Let (X, ΣX , µ) and (Y, ΣY , ν) be measure spaces, with measure algebras (B, µ ¯) and (A, ν¯). Let f : X → Y be inverse-measure preserving. Then we have a sequentially order-continuous measure-preserving Boolean homomorphism φf : A → B defined by setting φf F • = f −1 [F ]• for every F ∈ ΣY . Proof : Since f : X → Y is inverse-measure-preserving, f −1 [F ] ∈ ΣX and µ(f −1 [F ]) = νF for every F ∈ ΣY . Following Theorem 79 it is immediate that c 2018, Indian Institute of Technology Delhi
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φf sequentially order-continuous Boolean homomorphism. Using µ(f −1 [F ]) = νF for every F ∈ ΣY , we get µ ¯(f −1 [F ]• ) = µ ¯(φf F • ) = ν¯F • from Definition 77. It is proved that φf : A → B is sequentially order-continuous measure-preserving Boolean homomorphism. Table 4.13 summarizes all the arrows of base categories. Arrows of category C f • a.e. class of Non-singular(M0R) measurable f : X → Y f • a.e. class of Inverse-measure-preserving f : X → Y Non-singular(M0R) measurable f : X → Y Inverse-measure-preserving f : X → Y Partial Injections or Partial measurable maps
Arrows of category Cop Sequentially order-continuous Boolean homomorphism πf : B → A SOC measure preserving Boolean homomorphism πf : B → A Sequentially order-continuous Boolean homomorphism πf : B → A SOC measure preserving Boolean homomorphism πf : B → A Same as arrows of Inverse Category C
Table 4.13: The arrows of various base categories from which appropriate functors L0 , L2 , l0 , l2 are defined.
4.3.4
Codomain categories
All Lp function spaces are Riesz spaces; see [Fre16], [Fre12] for instance. This suggests that the generic codomain category for L0 and L2 (or Lp , 1 ≤ p ≤ ∞) functors is the category Riesz with objects as Riesz spaces and arrows as Riesz homomorphisms. However depending upon the kind of structure one is interested on signal space, we briefly tabulate different codomain categories with objects carrying those structures and corresponding structure-preserving morphisms as summarized in Table 4.14.
The standard categories of interest for L0 is the category of Riesz spaces and Riesz homorphisms (multiplicative sequentially order-continuous Riesz homomorphism); while for Lp is the category of normed Riesz spaces and norm-presereving Riesz homorphisms (multiplicative sequentially order-continuous Riesz homomorphism). For L2 the category of Hilbert spaces with continuous linear maps is often most important from application perspective. c 2018, Indian Institute of Technology Delhi
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Structure Linear space Partial order Lattice Partial ordered Linear Space Riesz Space Topological Linear Space Banach Lattice
120
Structure preserving morphism Linear map Monotone (order-preserving) map Lattice (order-continuous) homomorphism Positive (order-preserving) Linear map Riesz homomorphism Continuous linear map Bounded Riesz homomorphism
Table 4.14: The combination of structures in codomain categories for functors L0 , L2 , l0 , l2 .
4.4
On L0,L2 functors
Having studied the domain and codomain categories, in this section we formulate various possible major cases of L0 and L2 functors. The main references for these cases are volumes [Fre16], [Fre12] of the treatise on unified measure theory.
4.4.1
Results on functors L0
In this section, we specifically discuss all results pertaining to the usual L0 construction of measure theory which can be viewed appropriately as a functor. The discussion and proof for next result is adapted from Exercise 241X(g) of [Fre16].
Proposition 81 [Fre16] Let (X, ΣX , N (µ)) and (Y, ΣY , N (ν)) be localizable measurable spaces, and φ : X → Y a non-singular measurable function. Then (a)For every real valued ΣY -measurable function g defined on Y , gφ is ΣX -measurable. (b) The map g 7→ gφ : L0 (ν) → L0 (µ) induces T : L0 (ν) → L0 (µ) defined by setting T g • = (gφ)• for every g ∈ L0 . (c) T is linear operator, that preserves multiplicative structure T (v × w) = T v × T w for all v, w ∈ L0 (ν), and also lattice structure, that is T (supn∈N vn ) = supn∈N T vn given hvn in∈N is a sequence in L0 (ν) with a supremum in L0 (ν).
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Proof : (a) Noting that every inverse-measure-preserving function by its definition is always measurable and using Lemma 49, it is immediate that gφ is ΣX -measurable. Briefly, Let a ∈ R. There exists an F ∈ ΣY such that {y : g(y) ≤ a} = F . Again there exists an E ∈ ΣX such that φ−1 [F ] = E. Hence {x : gφ(x) ≤ a} = E ∈ ΣX . Since a is arbitrary, gφ is ΣX -measurable. (b) By definition, L0 (ν) is the set of real-valued functions f defined on conegligible subsets of Y which are virtually measurable, that is, such that f C is measurable for some conegligible set C ⊆ Y . In other words, L0 (ν) is just the set of real-valued functions f , defined on subsets of Y , which are equal almost everywhere to some ΣY measurable function g from Y to R. More precisely, if g : Y → R is ΣY -measurable and f =a.e. g, then C = {y : y ∈ dom f, f (y) = g(y)} is conegligible and f C = gC is measurable, so f ∈ L0 (ν) by part(h) of Theorem 50. Now by Proposition 52 =a.e. is an equivalence relation on L0 (ν) and the space L0 (ν) is the set of equivalence classes in L0 (ν) under =a.e. . For f ∈ L0 (ν), we denote f • for its equivalence class in L0 (ν). Consider f ∈ L0 (ν) and g ∈ L0 (ν) such that f • = g • in L0 (ν). Then C = {y : y ∈ dom f, f (y) = g(y)} is conegligible and f C = gC is measurable. Again repeating arguments of Lemma 49 we may conclude (f φ)• = (gφ)• . Explicitly let A = φ−1 [C] = dom(gC)φ, which is surely conegligible (since φ is non-singular or zero-reflecting) and a ∈ R. Since f C = gC is ΣY -measurable, by Definition 39 there exists an F ∈ ΣY such that {y : (gC)(y) ≤ a} = {y : (f C)(y) ≤ a} = F ∩ C. On the other hand, there exists an E ∈ ΣX such that φ−1 [F ] = E. Hence {x : (gC)φ(x) ≤ a} = {x : (f C)φ(x) ≤ a} = A ∩ E ∈ ΣA . Since a is arbitrary, (gC)φ = (f C)φ is ΣX -measurable. But gφ : X → R is ΣX -measurable and extends (gC)φ = gφA on a conegligible A ⊆ X. Consequently gφ =a.e. (gC)φ =a.e. (f C)φ and (f φ)• = (gφ)• in L0 (µ). Indeed we have well-defined T : L0 (ν) → L0 (µ) where T g • = (gφ)• for every g ∈ L0 (ν). (c)(i) By part (b) of Theorem 50 , g + g 0 is ΣY -measurable, where (g + g 0 )(y) = g(y) + g 0 (y) for y ∈ Y . Therefore there exists an F ∈ ΣY such that {y : (g + g 0 )(y) ≤ a} = {y : g(y) + g 0 (y) ≤ a} = F . But there exists an E ∈ ΣX such that φ−1 [F ] = E. Hence {x : (g + g 0 )φ(x) ≤ a} = {x : gφ(x) + g 0 φ(x) ≤ a} = E ∈ ΣX . Since a is arbitrary, gφ+g 0 φ = (g+g 0 )φ is ΣX -measurable. Now from part (b) of Proposition 56, (g + g 0 )• = g • + g 0• in L0 (ν) and (gφ)• + (g 0 φ)• = ((g + g 0 )φ)• in L0 (µ). Hence c 2018, Indian Institute of Technology Delhi
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T (g + g 0 )• = ((g + g 0 )φ)• = (gφ)• + (g 0 φ)• = T g • + T g 0• . Similarly using Theorem 50 and part (c) of Proposition 56 it follows in same manner for any scalar c ∈ R, that T (c.g)• = ((c.g)φ)• = c.(gφ)• = c.T g • . Hence T is a linear operator. (c)(ii) Using Theorem 50 and part (d) of Proposition 56 it follows, that T (g × g ) = ((g × g 0 )φ)• = (gφ)• × (g 0 φ)• = T g • × T g 0• . Hence T preserves multiplication. 0
•
(c)(iii) Using Theorem 50 and part (e) of Proposition 56 it follows , that T (supn∈N gn )• = ((supn∈N gn )φ)• = supn∈N (gφ)• = supn∈N T g • . Hence T preserves lattice structure. Proposition 81 proves that L0 construction is truly functorial; it explicitly sets up a recipe for defining L0 functor. Also note that we are not really making any use of second clause in the definition of inverse-measure preserving function, that is µ(φ−1 [F ]) = νF for every F ∈ ΣY . This is intimately related to the fact that L0 theory truly involves only Dedekind σ-complete Boolean algebras rather than measure algebras as Theorem 84 makes this precise. In other words, the measures enter theory only via their ideals of negligible sets. This directly influences the definition of L0 functor and stands as motivation behind the category of localizable measurable spaces and zero-reflecting maps apart from its good properties and generalization to partial monic derivatives. Next definition is explicitly adapted from the discussions in [Fre12].
Definition 82 ( [Fre16], [Fre12]) Let LocMeas, Riesz be the categories as defined earlier. The (contravariant) functor L0 : LocMeas → Riesz is defined as a mapping that associates to each measurable space (X, ΣX , N (µ)) an object L0 (X, ΣX , N (µ)) := {f • |f ∈ L0 (µ)}. The assignment of morphism φ• : (X, ΣX , N (µ)) → (Y, ΣY , N (ν)) is given by L0 (φ• )(g • ) = (gφ)• (4.14) where L0 (φ• ) : L0 (Y, ΣY , N (ν)) → L0 (X, ΣX , N (µ)) is the morphism in Riesz between Riesz spaces with multiplication, which is also multiplicative sequentially ordercontinuous. c 2018, Indian Institute of Technology Delhi
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We verify the functoriality of L0 now. The object L0 (X, ΣX , N (µ)) = L0 (µ) is a well-defined Archimedean and Dedekind σ-complete Riesz space; refer Sections 4.5.1, 4.5.2. Thus it is a well-defined object of category Riesz. Next we check if L0 (φ• ) is a well-defined morphism of Riesz. From Proposition 81, every g • is sent to L0 (φ• )(g • ) = (gφ)• which is a proper element of L0 (Y, ΣY , N (ν)) = L0 (ν) proving that L0 (φ• ) is a well-defined morphism in Riesz. Now we check if composition is preserved with the help of commutative composition diagram,
(R, ΣB ) O
gφ
(X, ΣX )
id gφ
φ
/ (R, ΣB ) 9 O f ψ=g
/
(Y, ΣY )
id fψ
ψ
/ (R, ΣB ) 9 O
(4.15)
f
/ (Z, ΣZ )
(L0 (φ• ) ◦ L0 (ψ • ))(f • ) = L0 (φ• )[L0 (ψ • )(f • )] = L0 (φ• )(f ψ)• = (f ψφ)•
= L0 (ψφ)• (f • ) = (L0 (ψ • · φ• ))(f • )
(4.16)
(4.17)
Finally for the identity, L0 (id•(Y,ΣY ,N (ν)) )(g • ) = (g·id)• = g • implying L0 (id•(Y,ΣY ,N (ν)) ) = idL0 (Y,ΣY ,N (ν)) . Indeed functor L0 is a well-defined contravariant functor. Depending upon the particular structures or their combinations such as vector, partial order, lattice and multiplication one can easily form variations of Definition 82 which is a sort of prototype, using different codomain categories as shown in Table 4.14. One could also generalize to complex L0 using spaces based on complexvalued functions instead of real-valued functions bearing in mind that the usual order structure does not hold. Here one can make use of L0 C = L0 C (ν) for the space of complex-valued functions g such that dom g is a conegligible subset of Y and there is a conegligible subset C ⊆ Y such that gC is measurable; which means the real and imaginary parts of g both belong to L0 (ν). c 2018, Indian Institute of Technology Delhi
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In the special case of µ and ν being counting measures (and therefore pointsupported by Definition 35) on sets X and Y the Riesz spaces L0 (µ) becomes L0 (µ) = RX itself, since ΣX = PX and every set except the empty set has a non-zero measure or in other words the null ideal is trivial. Hence we restrict to a subcategory countLocMeas of LocMeas with objects of type (X, PX, Φ) where Φ is trivial null ideal with single element the usual empty set. Also note that here equivalence class φ• coincides with the measurable map φ itself, since we quotient by trivial null ideal.
Definition 83 ( [Fre16], [Fre12]) Let countLocMeas, Riesz be the categories as defined earlier. The (contravariant) functor l0 : countLocMeas → Riesz is defined as a mapping that associates to each measurable space (X, PX, Φ) an object l0 (X, PX, Φ) := {f ∈ L0 (µ)}. The assignment of morphism φ : (X, PX, Φ) → (Y, PY, Φ) is given by l0 (φ)(g) = gφ (4.18) where l0 (φ) : l0 (Y, PY, Φ) → l0 (X, PX, Φ) is the morphism in Riesz between Riesz spaces with multiplication, which is also multiplicative sequentially order-continuous.
We now state the dual of Proposition 81 which leads to a covariant form of L0 from opposite of LocMeas which we denote as compBoolAlg. Note that in general these categories are not equivalent. Theorem 84 (364P [Fre12]) If B and A are Dedekind σ-complete Boolean algebras, and π : B → A is a sequentially order-continuous Boolean homomorphism, then (a) We have a multiplicative sequentially order-continuous Riesz homomorphism Tπ : L0 (B) → L0 (A) defined by the formula [[Tπ u > a]] = π[[u > a]] whenever a ∈ R and u ∈ L0 (B). (b) Tπ is order-continuous iff π is order-continuous, injective iff π is injective, surjective iff π is surjective.
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(c) If C is another Dedekind σ-complete Boolean algebra and θ : A → C another sequentially order-continuous Boolean homomorphism then Tθπ = Tθ Tπ : L0 (B) → L0 (C). The proof of theorem can be found in [Fre12]. If B is the Boolean algebra quotient ΣX /ΣX ∩ N (µ) then L0 (µ) gets identified with L0 (B). Correspondingly each g • ∈ L0 (µ) is identified with u ∈ L0 (B) which is a 7→ [[g • > a]] : R → B, for every a ∈ R. Now [[g • ∈ E]] means the region where g • takes values in E or [[g • ∈ E]] = g −1 [E]• . Hence [[g • > a]] = [[g • ∈ (a, ∞) ]] or ξ • : E • 7→ g −1 [E]• for any Borel set E ⊆ R. In conclusion u or g • corresponds to Boolean homomorphism ξ : (ΣB /(ΣB ∩ N )) → B. This theorem therefore gives rise to an explicit Definition 85. Definition 85 [Fre12] Let compBoolAlg, Riesz be the categories as defined earlier. The (covariant) functor L0 : compBoolAlg → Riesz is defined as a mapping that associates to each Dedekind σ-complete Boolean algebra B an object L0 (B) := {u |a 7→ [[u > a]]for every a ∈ R}. The assignment of morphism π : B → A is given by L0 (π)(u) defined by [[L0 (π)u > a]] = π[[u > a]] (4.19) where L0 (π) : L0 (B) → L0 (A) is the morphism in Riesz between Riesz spaces with multiplication, which is also multiplicative sequentially order-continuous.
4.4.2
Results on functors L2
Analogous to Proposition 81, Proposition 86 proves that L2 construction is really functorial; it explicitly sets up a recipe for defining L2 functor. The primary difference is that we now make use of second clause in the definition of inverse-measure preserving function, that is µ(φ−1 [F ]) = νF for every F ∈ ΣY . This is again intimately related to the fact that L2 theory truly involves measure spaces or dually measure algebras and normed Riesz spaces as Proposition 86 makes this precise. Proposition 86 (244Xo [Fre16]) Let (X, Σ, µ) and (Y, T, ν) be measure spaces, and φ : X → Y an inverse-measure-preserving function. Then, (a)For every [−∞, ∞]-real c 2018, Indian Institute of Technology Delhi
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valued ΣY -measurable function g defined on Y , gφ is ΣX -measurable. (b) The map g 7→ gφ : L2 (ν) → L2 (µ) induces T : L2 (ν) → L2 (µ) defined by setting T g • = (gφ)• for every g ∈ L2 such that |g|2 is integrable (often termed square integrable functions) (c) T is norm-preserving linear operator that is, kT g • k2 = kg • k2 for every g • ∈ L2 (ν), which preserves multiplicative structure T (v ×w) = T v ×T w for all v, w ∈ L2 (ν), and also lattice structure, that is T (supn∈N vn ) = supn∈N T vn given hvn in∈N is a sequence in L2 (ν) with a supremum in L2 (ν). Proof : By definition, L2 = L2 (µ) is simply the set of functions g ∈ L0 = L0 (µ) such that |g|2 is integrable, and L2 = L2 (µ) is defined as the set of functions {g • : g ∈ L2 } ⊆ L0 = L0 (µ). Also L2 (µ) is a Riesz subspace of L0 (µ). In Proposition 81, we have already proved that Tφ : L0 (ν) → L0 (µ) is a Riesz homomorphism, that is, a linear operator which preserves both multiplicative and lattice structure. Since here T : L2 (ν) → L2 (µ) is really a restriction of Tφ operating on equivalence classes g • with g ∈ L0 = L0 (µ) such that |g|2 is integrable. Hence T : L2 (ν) → L2 (µ) is also a Riesz homomorphism and therefore a linear operator which preserves both multiplicative and lattice structure. All now remains to be proven is that, it is also norm preserving. For this recall that k k2 : L2 = L2 (µ) → [0, ∞[ is actually a norm defined by R writing kg • k2 = kgk2 for every g ∈ L2 where kgk2 = ( |g|2 )1/2 . Now a real-valued R |g|2 = f is integrable means it is expressible as f 0 − f 00 with integral as f = R 0 R 00 f − f . where f 0 and f 00 are such that: (i) the domain of f 0 is a conegligible subset of X and f 0 (x) ∈ [0, ∞[ for each x ∈ dom f 0 (and same for f 00 ), (ii) there exists a non-decreasing sequence hfn0 in∈N of non-negative simple functions such that R supn∈N fn0 < ∞ and limn→∞ fn0 (x) = f 0 (x) for almost every x ∈ X (and same for R R f 00 ). Hence f 0 = limn→∞ fn0 whenever hfn0 in∈N is a non-decreasing sequence of simple functions converging to f 0 almost everywhere. Now for simplicity we assume |g|2 = f is a simple function and prove for this case, but general case is not much difficult since it simply involves taking supremum of non-decreasing sequence of simple R R P functions as described earlier. In this special case, |g|2 = f = m i=0 ai νFi where Pm f = i=0 ai χFi and every Fi ∈ ΣY is a measurable set of finite measure. But |g|2 R P being simple means g is also simple and ai = |bi |2 . Hence g = m i=0 bi νFi where c 2018, Indian Institute of Technology Delhi
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P Pm g= m i=0 bi χFi . Now gφ = ( i=0 bi χFi )φ, but χFi φ = χEi since there exist Ei ∈ ΣX Pm Pm −1 2 such that φ [Fi ] = Ei . Hence gφ = i=0 bi χEi and |gφ| = i=0 ai χEi . Now using crucial property of inverse-measure-preserving, that is µEi = µ(φ−1 [Fi ]) = νFi , R R 2 Pm Pm |gφ|2 = |g| . Therefore kgφk2 = kgk2 implying i=0 ai µEi = i=0 ai νFi = • • kT g k2 = kg k2 . Indeed generalizing to non-simple integrable g, it is proved that T : L2 (ν) → L2 (µ) is a norm preserving Riesz homomorphism. The spaces Lp (X, ΣX , µ) for any p ∈ [1, ∞] are Banach Lattices (see [Fre12]), hence instead of considering simply Riesz the category of Riesz spaces and Riesz homomorphisms, one can consider a codomain category BanLatt the category of Banach Lattices (complete normed Riesz spaces) and Riesz homomorphisms (ordercontinuous Positive operators) which are contractions. The objects additionally also have the property of multiplication and therefore homomorphisms become multiplicative when they preserve this property. Definition 87 ( [Fre16], [Fre12]) Let LocMeasure, BanLatt be the categories as defined earlier. The (contravariant) functor L2 : LocMeasure → BanLatt is defined as a mapping that associates to each measure space (X, ΣX , µ) an object L2 (X, ΣX , µ) := {f • |f ∈ L0 (µ), |f |2 is integrable}. The assignment of inversemeasure -preserving morphism φ : (X, ΣX , µ) → (Y, ΣY , ν) is given by L2 (φ)(g • ) = (gφ)•
(4.20)
where L2 (φ) : L2 (Y, ΣY , ν) → L2 (X, ΣX , µ) is the morphism in BanLatt between Banach lattices with multiplication, which is also multiplicative norm-preserving, that is kL2 (φ)g • k2 = kg • k2 . The functoriality of L2 easily gets verified in the same manner as done for Definition 82 of L0 functor. Once again depending upon the particular structures or their combinations such as vector, partial order, lattice, multiplication and complete norm one can easily form variations of Definition 87 which is a sort of prototype, using different codomain categories. c 2018, Indian Institute of Technology Delhi
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From viewpoint of signal representation, the most common variation is to consider structure of completed normed vector space or Banach space and generalize to complex L2 using spaces based on complex-valued functions. Again we need to consider L0 C = L0 C (ν) for the space of complex-valued functions g that are virtually measurable. In other words, L0 C (ν) is just the set of complex-valued functions f , defined on subsets of X, which are equal almost everywhere to some ΣX -measurable function h from X to C. Definition 88 ( [Fre16], [Fre12]) Let Measure, Hilb be the categories as defined earlier. The (contravariant) functor L2 : Measure → Hilb is defined as a mapping that associates to each measure space (X, ΣX , µ) an object L2 (X, ΣX , µ) := {f • |f ∈ R L0C (µ), |f |2 is integrable} := {h : X → C | |h(x)|2 dµ < ∞}. The assignment of inverse-measure -preserving morphism φ : (X, ΣX , µ) → (Y, ΣY , ν) is given by L2 (φ)(g • ) = (gφ)•
(4.21)
where L2 (φ) : L2 (Y, ΣY , ν) → L2 (X, ΣX , µ) is the morphism in Hilb between Hilbert spaces which is also norm-preserving, that is kL2 (φ)g • k2 = kg • k2 . Again in the special case of counting measures on sets X and Y the Hilbert spaces L (µ) becomes L2 (µ) itself and customarily denoted as l2 (X) since ΣX = PX and every set except the empty set has a non-zero measure or in other words the null ideal is trivial. Hence we restrict to a subcategory countMeasure of Measure with objects of type (X, PX, counting) where counting is point-supported counting measure. 2
Definition 89 ( [Fre16], [Fre12]) Let countMeasure, Hilb be the categories as defined earlier. The (contravariant) functor l2 : countMeasure → Hilb is defined as a mapping that associates to each measure space (X, PX, counting) an object P l2 (X, PX, counting) := {f ∈ L0C (µ), |f |2 is integrable} := {f : X → C | x∈X |f (x)|2 < ∞}. The assignment of inverse-measure -preserving morphism φ : (X, PX, counting) → (X, PY, counting) is given by l2 (φ)(g) = (gφ) c 2018, Indian Institute of Technology Delhi
(4.22)
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where l2 (φ) : l2 (Y, PY, counting) → l2 (X, PX, counting) is the morphism in Hilb between Hilbert spaces which is also norm-preserving, that is kl2 (φ)gk2 = kgk2 .
Before we can handle the dual of Proposition 86 which leads to a covariant form of L from opposite of LocMeasure which we denote as MeasureAlg, it is important to note that if B is any algebra carrying two non-isomorphic measures say µ and µ0 , the corresponding L2 (µ) and L2 (µ0 ) spaces are still isomorphic. Hence rather than defining the functor from MeasureAlg often functor LP (1 ≤ p < ∞) is determined by Boolean ring of elements of finite measure in a measure algebra than in terms of the whole algebra. Note that the algebra B is uniquely determined in certain cases but the measure µ ¯ is never determined. If (B, µ ¯) is a measure algebra, then a f Boolean ring is the ideal B = {b : b ∈ B, µ ¯b < ∞} whereas a ring homomorphism f f π : B → A is termed as measure-preserving if ν¯πb = µ ¯b for every b ∈ Bf . Theorem 90 makes this dependency precise. 2
Theorem 90 (366H [Fre12]) If (B, µ ¯) and (A, ν¯) are measure algebras and π : Bf → Af a measure-preserving ring homomorphism between corresponding rings of elements of finite measure, then (a) There is a unique order-continuous norm-preserving Riesz homomorphism Tπ : L2 (B, µ ¯) → L2 (A, ν¯) such that [[Tπ u > a]] = π[[u > a]], kTπ uk2 = kuk2 for every u ∈ L2 (B, µ ¯) and a > 0. ¯ is another measure algebra and θ : Af → Cf another measure(b) If (C, λ) ¯ preserving ring homomorphism, then Tθπ = Tθ Tπ : L2 (B, µ ¯) → L2 (C, λ).
For proof pertaining to general p see [Fre12]. This theorem therefore gives rise to an explicit Definition 91.
Definition 91 ( [Fre12]) Let MeasureAlg, BanLatt be the categories as defined earlier. The (covariant) functor L2 : MeasAlg → BanLatt is defined as a mapping that associates to each measure algebra (B, µ ¯) an object L2 (B, µ ¯) := {u |a 7→ c 2018, Indian Institute of Technology Delhi
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[[u > a]]for every a ∈ R}. The assignment of morphism π : (B, µ ¯) → (A, ν¯) is given 2 by L (π)(u) defined by [[L2 (π)u > a]] = π 0 [[u > a]]; π 0 : Bf → Af , a > 0
(4.23)
where L2 (π) : L2 (B, µ ¯) → L2 (A, ν¯) is the morphism in BanLatt between Banach lattices with multiplication, which is also multiplicative norm-preserving, that is kL2 (π)uk2 = kuk2 . As expected, L2 (π) : L2 (B, µ ¯) → L2 (A, ν¯) corresponds to the map g • 7→ (gφ)• : L1 (ν) → L1 (µ) of Theorem 86, when φ : X → Y is an inverse-measure-preserving function between (X, Σ, µ) and (Y, T, ν) with π : (B, µ ¯) → (A, ν¯) the corresponding measure-preserving homomorphism. Finally Table 4.15 summarizes the essential functorial definitions and their dual versions which we considered in detail here.
4.4.3
Generalization to partial categories
For a reader purely interested in signal representation and redundacy from a category theory viewpoint this section can be omitted. The reason we are including it here is because these extensions were motivated by [Heu13] and [Dmi11] earlier when we constructed a visual proof of concept. However later we realized that unified treatment of measure theory in [Fre11], [Fre16], [Fre12], [Fre13] has studied the functorial aspects of L0 and Lp constructions especially from the categories of Boolean and measure algebras. This made our task easier while at the same time we could connect these works developing a much broader perspective and emphasizing the functorial aspects from the application viewpoint. The category LocMeas being equivalent to the opposite of category of commutative von Neumann algebras [Dmi11] has desirable properties with all finite transversal c 2018, Indian Institute of Technology Delhi
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Contravariant Functorial Model (R, Σ ) O B gφ
(I, ΣI )
/
id gφ
(R, Σ ) 9 O B g
/
φ
L0 (I, ΣI , N (µ)) o
L2 (µ) o
φ•
φ•
/
πξ
w
πξ
L0 (J, ΣJ , N (ν))
L0 (B) o
/ (J, ΣJ , N (ν))
(ΣI /(ΣI ∩ N (µ))) o
π
(ΣJ , 4, ∩)
πφ
L2 (B, µ ¯) o
(J, ΣJ , ν)
Bf o
L2 (πφ )
(B, µ ¯) o
ξ(E7→g −1 [E])
L0 (B)
L0 (πφ )
L2 (ν)
L0 : LocMeas → Riesz L2 : LocMeasure → BanLatt
(ΣB , 4, ∩)
id
(ΣI , 4, ∩) o
T =L2 (φ• )
(I, ΣI , µ)
(ΣB , 4, ∩) o
(J, ΣJ )
T =L0 (φ• )
(I, ΣI , N (µ))
Dual Covariant Functorial Model
(ΣJ /(ΣJ ∩ N (ν))) L2 (A, ν¯)
π φ |f
πφ
Af
(A, ν¯)
L0 : compBoolAlg → Riesz. L2 : MeasureAlg → BanLatt
Table 4.15: Duality between covariant and contravariant forms of L0 and L2 summarized. limits which makes it possible to consider its monic and partial derivatives. Being in possession of a category with limits such as equalizers, products and therefore pullbacks, one can immediately form the monic and partial derived categories from the original category [CL02]. First we consider the M-category of LocMeas. The M-category (LocMeas, M) of (stable) monics in the category LocMeas consists of • objects the collection (X, ΣX , N (µ)), (Y, ΣY , N (ν)), ... ∈ Ob(LocMeas) • morphisms for a pair X, Y ∈ Ob(LocMeas), LocMeas(X, Y) = {f : X → Y | f is monic }.
Since the isomorphisms in parent category are monic they get included in this category. Further pullbacks in LocMeas of monic along general map f exists and is c 2018, Indian Institute of Technology Delhi
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itself a monic (hence an M-morphism). m0
X ×Y Z /
/
Z
f0
f
X /
m
/
Y
A general M-category (C, M) is also sometimes denoted as (C, M onic). An easier example is (Set, M) or (Set, M onic) which is the category of sets with all injections. Next consider a partial (restriction) category Par(LocMeas, M) of the above M-category which consists of: • objects: a collection X, Y, Z... ∈ Ob(LocMeas) • morphisms: pair (m, f ) : X → Y (upto equivalence) where m ∈ (LocMeas, M) 0 and f ∈ LocMeas: ~X m
f
~
X
Y
• identity: (1X , 1X ) : X → X • composition: (m0 , g)(m, f ) = (mm00 , gf 0 ):
00 X ~
m00
~X m
0
f0
~
0
m0
f
~
X
}Y
} !
Y
g
Z
• Restriction: for the morphism (m, f ) : X → Y, (m, f ) = (m, m) : X → X with the usual unit and associative laws. Up-to equivalence means we factor out by the equivalence relation (∼) where (m, f ) ∼ (m0 , f 0 ) if there exists an isomorphism g : X0 → X00 such that m0 ◦ g = m and f 0 ◦ g = f .
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Here a basic example to keep in mind is Par(Set, M) which is the category of sets and partial maps. Similarly Par((Set, M), M) is the category PInj of sets and partial injections. Further Par(LocMeas, M) is the category of localizable measurable spaces and partial measurable maps. Par((LocMeas, M), M) is the category of localizable measurable spaces and partial monic measurable maps Categories of partial maps were given the characterization of restriction structure and categories by [CL02]. They are widely seen as a simple equational axiomatization for categories of partial maps. In-fact every restriction category embeds fully and faithfully into a partial map category. Very brief definitions of the restriction and dagger category are recalled in A.1. Using these of these, Proposition 92 is immediate for the category studied in this subsection.
Proposition 92 Every inverse category is a dagger category. In particular Par((LocMeas, M), M) is a dagger category.
Proof : From the definitions of inverse and dagger categories we immediately deduce that dagger structure is given by †(1X ) = 1X for all objects X and f † = f ◦ for all morphisms f in the inverse category C. It is clear that Par((LocMeas, M), M) being an inverse category is also dagger, which implies that every partial monic measurable non-singular morphism can be reversed uniquely which we denote as its adjoint morphism. Finally we are ready to extend the basic L0 functor to the domain category constructed here.
4.4.4
The L0 functor extended to partial category
In this section we briefly touch upon the well-known case of functor l2 : PInjop → Hilb which was first observed by [Bar92], developed in [Heu13]. This section is not essential purely from the perspective of signal representation however we studied this
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case and tried generalizing it to measure spaces from sets in reverse fashion. We show how it can be recast as the special case of Definition 93.
Definition 93 Let Par(LocMeas, M) ,Riesz be the categories. The functor L0 : Par(LocMeas, M) → Riesz is defined as a mapping that associates to each measurable space (Y, ΣY , N (ν)) an object L0 (Y, ΣY , N (ν)) := {g • |g ∈ L0 (ν)}. The assignment of morphism f • : X → Y or (X o f1 o W f2 / Y) is given by L0 (f • )(g • ) = (gf )• orL0 (f • )(g • )(x) = g • (f1 (w))
(4.24)
where L0 (f • ) : L0 (Y) → L0 (X) is the morphism in Riesz.
4.4.5
The case of linear Borel measurable function
Propositions 81 and 86 can be used for non-singular and inverse-measure-preserving maps between local measurable spaces with identity map on (R, ΣB ). However for mathematical modeling of transformations involving change in signal amplitudes we need a general Borel measurable function h : (R, ΣB ) → (R, ΣB ). Although this ¯ : L0 (µ) → L0 (µ) it is not necessarily linear transinduces in general a function h formation. If we need this endo-transformation to be invertible and linear (which becomes an arrow in category Hilb) we restrict to linear h which covers uniform amplitude scaling most frequent in image signals. The next proposition which we extend from exercise 241X(h) [Fre16] makes this precise.
Proposition 94 If (X, Σ, µ) is a measure space and h : R → R is a Borel measurable function. Then hf ∈ L0 = L0 (µ) for every f ∈ L0 and hf =a.e. hg whenever f =a.e. g. ¯ : L0 → L0 by setting h(f ¯ • ) = (hf )• for every f ∈ L0 ; then h ¯ Define a function h becomes a linear operator when h is a linear function.
Proof : Using parts (f) and (g) of Theorem 50 and Proposition 56 it follows that hf ∈ L0 = L0 (µ) for every f ∈ L0 .
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For simplicity we assume that f : X → R is ΣX -measurable while h : R → R is ΣB -measurable or more precisely dom h = R and dom f = X f =a.e. g, then C = {x : x ∈ dom g, f (x) = g(x)} is conegligible and f C = gC is measurable. Now if a ∈ R, then {y : h(y) < a} = E, where E is a Borel subset of R and (f C)−1 [E] is of the form F ∩ dom(f C), where F ∈ ΣX , then {x : (h(f C))(x) < a} = {x : (h(gC))(x) < a} = F ∩ C ∈ ΣC . As a is arbitrary, h(f C) = h(gC) is ΣX -measurable. But hf : X → R is ΣX -measurable and extends h(f C) = hf C on a conegligible C ⊆ X. Consequently hf =a.e. h(f C) =a.e. h(gC) and (hf )• = (hg)• ¯ : L0 (µ) → L0 (µ) where hf ¯ • = (hf )• for every in L0 (µ). Indeed we have well-defined h f ∈ L0 (µ). Next by part (b) of Theorem 50, f + f 0 is ΣX -measurable, where (f + f 0 )(x) = g(x) + g 0 (x) for x ∈ X. Therefore there exists an F ∈ ΣX such that {x : (f + f 0 )(x) < a} = {x : f (x) + f 0 (x) < a} = F . But there exists a Borel B = (∞, a0 ] ∈ ΣB such that h−1 [B] = E = (∞, a] and f −1 [E] = F ∈ ΣX . Hence {x : [h(f + f 0 )](x) ≤ a0 } = {x : h(f (x) + f 0 (x)) ≤ a0 } = F ∈ ΣX . Since a0 is arbitrary, hf + hf 0 = h(f + f 0 ) is ΣX -measurable. Now from part (b) of Proposition 56 (f +f 0 )• = f • +f 0• in L0 (µ) and using the fact h(y + y 0 ) = hy + hy 0 for a linear h we have (h(f + f 0 ))• = (hf + hf 0 )• = ¯ + f 0 )• = (h(f + f 0 ))• = (hf )• + (hf 0 )• = hf ¯ • + hf ¯ 0• . (hf )• + (hf 0 )• in L0 (µ). Hence h(f Similarly using Theorem 50 and part (c) of Proposition 56 it follows in same manner ¯ ¯ • . Hence h ¯ is a linear for any scalar c ∈ R, that h(c.f )• = (h(c.f ))• = c.(hf )• = c.hf operator.
4.5
Functorial signal space properties
In this section, we discuss differences between classic versus functorial signal space properties. This includes standard properties such as linearity, lattice and multiplication of classical signal spaces L0 (R, ΣLeb , N (µ) (or L0X ) along with additional norm and inner-product of L2 (R, ΣLeb , µ). In general all these properties in subcategory of Riesz or BanLatt are simply local or valid on objects within this category. The structure-preserving morphisms in these subcategories are linear operators preserving these properties considered as structures on objects. References for these properties c 2018, Indian Institute of Technology Delhi
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are [Fre16], [Fre12].
4.5.1
Linear structure
For the spaces given by Definitions 53, 54, 57,58, we have the following properties, 1. Let f , f 0 , g, g 0 ∈ L0 , f =a.e. f 0 and g =a.e. g 0 then f + g =a.e. f 0 + g 0 . This defines addition on L0X a subspace with f , f 0 , g, g 0 defined on complete X. Also it defines addition on L0 (µ) by setting f • + g • = (f + g)• for all f , g ∈ L0 . Let f , g ∈ L2 . If c, c0 ∈ R then |c + c0 |2 ≤ 22 max(|c|2 , |c0 |2 ), therefore |f + g|2 ≤a.e. 22 (|f |2 ∨ |g|2 ). But |f + g|2 ∈ L0 and 22 (|f |2 ∨ |g|2 ) is integrable so |f + g|2 is integrable. Hence f + g ∈ L2 for all f , g ∈ L2 ; therefore f • + g • = (f + g)• for all f , g ∈ L2 defining an addition on L2 (µ). 2. If f , g ∈ L0 and f =a.e. g, then cf =a.e. cg for every c ∈ R. This defines scalar multiplication on L0X with f , g defined on complete X. And it defines scalar multiplication on L0 (µ) by setting c · f • = (cf )• for all f ∈ L0 and every c ∈ R. Now let f ∈ L2 and c ∈ R then |cf |2 = |c|2 |f |2 is integrable, so cf ∈ L2 . Therefore cf • ∈ L2 (µ) whenever f • ∈ L2 (µ) and c ∈ R. Thus L0X , L0 (µ) are linear spaces over R, with zero 0 (the function with domain X and constant value 0) and 0• respectively. The negatives are given by −(f ) and −(f • ) = (−f )• respectively. Also L2 (µ) is a linear subspace of L0 (µ). Thus all axioms of a linear space are satisfied namely, • f +(g +h) = (f +g)+h for all f , g, h ∈ L0 , hence f 0 +(g 0 +h0 ) = (f 0 +g 0 )+h0 for all f 0 , g 0 , h0 ∈ L0X . and f • + (g • + h• ) = (f • + g • ) + h• for all f • , g • , h• ∈ L0 (µ). • f + 0 = 0 + f = f for every f ∈ L0 , hence f 0 + 0 = 0 + f 0 = f 0 for every f 0 ∈ L0X , and f • + 0• = 0• + f • = f • for every f • ∈ L0 (µ). • f + (−f ) =a.e. 0 for every f ∈ L0 , hence f 0 + (−f 0 ) =a.e. 0 for every f 0 ∈ L0X , and f • + (−f )• = 0• for every f • ∈ L0 (µ). • f + g = g + f for all f , g ∈ L0 , hence f 0 + g 0 = g 0 + f 0 for all f 0 , g 0 ∈ L0X , and f • + g • = g • + f • for all f • , g • ∈ L0 (µ).
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• c(f + g) = cf + cg for all f , g ∈ L0 and c ∈ R, hence c(f 0 + g 0 ) = cf 0 + cg 0 for all f 0 , g 0 ∈ L0X and c ∈ R, and c(f • + g • ) = cf • + cg • for all f • , g • ∈ L0 (µ) and c ∈ R. • (c + c0 )f = cf + c0 f for all f ∈ L0 and c,c0 ∈ R, hence (c + c0 )g = cf + c0 g for all g ∈ L0X and c,c0 ∈ R, and (c + c0 )f • = cf • + c0 f • for all f ∈ L0 (µ) and c, c0 ∈ R. • (cc0 )f = c(c0 f ) for all f ∈ L0 and c, c0 ∈ R, hence (cc0 )g = c(c0 g) for all g ∈ L0X and c, c0 ∈ R, and (cc0 )f • = c(c0 f • ) for all f • ∈ L0 (µ) and c, c0 ∈ R. • 1f = f for all f ∈ L0 , hence 1f 0 = f 0 for all f 0 ∈ L0X , and 1f • = f • for all f • ∈ L0 (µ).
4.5.2
Partial order and lattice structure
For the spaces given by Definitions 53, 54, 57,58, we have properties related to partial order and lattice in addition to linearity. 1. Let f , f 0 , g, g 0 ∈ L0 , f =a.e. f 0 , g =a.e. g 0 and f ≤a.e. g, then f 0 ≤a.e. g 0 . This defines a relation ≤ on L0X by declaring f1 ≤ g1 iff f ≤a.e. g where f =a.e. f1 and g =a.e. g1 for f1 , g1 ∈ L0X which simply means f1 ≤a.e. g1 for f1 , g1 defined on whole domain X. Also it defines a relation ≤ on L0 (µ) by declaring that f • ≤ g • iff f ≤a.e. g. For p ∈ [1, ∞], f , g ∈ Lp means f , g ∈ L0 such that |f |p , |g|p are integrable thus ≤a.e. inherited from L0 defines a relation ≤ on Lp (µ) by saying that f • ≤ g • iff f ≤a.e. g for f , g ∈ Lp . Alternately we can inherit this relation directly from L0 (µ) by noting that Lp (µ) is its linear subspace. This of-course holds for special case p = 2. 2. Next ≤ is a partial order on L0X ,L0 (µ) and Lp (µ). (i) Let f , g, h ∈ L0 , if f ≤a.e. g and g ≤a.e. h, then f ≤a.e. h. Therefore f1 ≤ h1 whenever f1 , g1 , h1 ∈ L0X , f1 ≤ g1 and g1 ≤ h1 . Similarly f • ≤ h• whenever f • , g • , h• ∈ L0 (µ), f • ≤ g • and g • ≤ h• . (ii) Let f ∈ L0 and f ≤a.e. f ; then f1 ≤ f1 , f • ≤ f • for every f • ∈ L0 (µ). (iii) Let f , g ∈ L0 , f ≤a.e. g and g ≤a.e. f , then f =a.e. g, therefore if f1 ≤ g1 and g1 ≤ f1 then f1 = g1 in ∈ L0X , similarly f • ≤ g • and g • ≤ f • then f • = g • in L0 (µ). For linear subspace Lp (µ), ≤ is a partial order on inherited from L0 (µ) and holds for L2 (µ). 3. Adding linear properties of L0X , L0 (µ) and Lp (µ), with ≤, make these partially ordered linear spaces, which are real linear spaces with a partial order ≤ c 2018, Indian Institute of Technology Delhi
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such that for elements x, y, z of these we have (i) if x ≤ y then x + z ≤ y + z for every z, since if f , g, h ∈ L0 and f ≤a.e. g, then f + h ≤a.e. g + h. (ii) if 0 ≤ x then 0 ≤ cx for every c ≥ 0, since if f ∈ L0 and f ≥ 0 a.e., then cf ≥ 0 a.e. for every c ≥ 0. 4. Next L0X , L0 (µ) and Lp (µ) are Riesz spaces or vector lattices, which are partially ordered linear spaces such that x ∨ y = sup{x, y} and x ∧ y = inf{x, y} are defined for all x, y ∈ L0X , L0 (µ) or Lp (µ) Let f , g ∈ L0 such that f • = x and g • = y. Then f ∨ g, f ∧ g ∈ L0 , hence f1 ∨ g1 , for f1 ∧ g1 ∈ L0X . Now expressing (f ∨ g)(x) = max(f (x), g(x)), (f ∧ g)(x) = min(f (x), g(x)) for x ∈ (dom f ∩ dom g). But, for any h ∈ L0 , we have f ∨ g ≤a.e. h ⇐⇒ f ≤a.e. h and g ≤a.e. h, h ≤a.e. f ∧ g ⇐⇒ h ≤a.e. f and h ≤a.e. g, Hence for any z ∈ L0 (µ) we have (f ∨ g)• ≤ z ⇐⇒ x ≤ z and y ≤ z, z ≤ (f ∧ g)• ⇐⇒ z ≤ x and z ≤ y. Indeed (f ∨ g)• = sup{x, y} = x ∨ y, (f ∧ g)• = inf{x, y} = x ∧ y in L0 (µ). Again this lattice structure in linear subspace Lp (µ) is inherited from L0 (µ) and present in L2 (µ). Finally there are some additional properties of Archimedeaness and Dedekind completeness that can be found in [Fre16].
4.5.3
Multiplicative structure
For the spaces given by Definitions 53, 54, 57,58, we have properties related to multiplicative structure in addition to being Riesz spaces. Let f , f 0 , g, g 0 ∈ L0 , f =a.e. f 0 and g =a.e. g 0 then f ×g =a.e. f 0 ×g 0 . Thus multiplication in L0X is the usual multiplication from L0 , while the multiplication on L0 (µ) is defined by setting f • ×g • = (f ×g)• for all f , g ∈ L0 . This also valid for the subspace Lp (µ). For all x, y, z ∈ L0 (µ) and c ∈ R, the following properties can be easily verified, • x × (y × z) = (x × y) × z, • x × 1• = 1• × x = x, where 1 is the equivalence class of the function with constant value 1, • c(x × y) = cx × y = x × cy, c 2018, Indian Institute of Technology Delhi
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• x × (y + z) = (x × y) + (x × z), • (x + y) × z = (x × z) + (y × z), • x × y = y × x, • |x × y| = |x| × |y|, • x × y = 0 iff |x| ∧ |y| = 0, • |x| ≤ |y| iff there is a z such that |x| ≤ 1• and x = y × z.
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Chapter 5
Some applications of transformation categories
In this Chapter we visually describe how the concept of category got associated with leyton’s Generative theory [Ley01]. The chapter can also be thought of as spin-off of the main work in Chapters 3 4. Using the approach of visual group theory [Car09] we explore the connections between group action and category action. Then we explore base structured categories with its application in symmetry. We attempt modeling Leyton’s generative theory through base structured categories showing comparative analysis with original wreath group approach of author. Next we explore hierarchy of structures using preliminary n-category theory (see [BM10]). Then we try to formulate the generative geometry intuition in terms of base-structured categories. Finally we present a heuristic discussion on multi-object species of structures exploring the crucial intuitive concept of base point. The work here is far from being complete requiring manifold theory from a category viewpoint which we are in process of learning and exploring in the context of ideas presented in this chapter.
5.1
From group action to category action
Since this chapter exclusively deals with the characterization of a functor as a category action, we begin with a simple group action and illustrate using the intuitive visual approach of [Car09], the notion of a category action through visual diagrams. Recall that there are two different but equivalent definitions of a simple group action. First is the concrete, set-theoretic (object-based) definition of a group action;
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Definition 95 [MB99] Let X be a set and G be a group then a (left) group action is a function µ : G × X → X such that 1. µ(g2 , µ(g1 , x)) = µ(g2 g1 , x) 2. µ(id, x) = x
The second definition is abstract or arrow-based that defines same action as a permutation representation. This is an authentic arrow definition that follows the common arrow philosophy of category theory.
Definition 96 [MB99] A permutation representation of G on X is a homomorphism from G into S(X) where S(X) is the symmetry or automorphism group of X.
The following Lemma 97 establishes the equivalence between these definitions. Lemma 97 [MB99] If G is a group and X is a set then, 1. If µ : G × X → X is a group action then the map φ : G 7→ S(X) defined as φ(g)(x) = µ(g, x) is a permutation representation of G on S(X). 2. If φ : G 7→ S(X) is a permutation representation then a group action of G on X as is defined as µ(g, x) := φ(g)(x).
The proof is left to the reader and can be found in [?]. Now observe that the group homomorphism is same as functor F : G → Set where G is a category with a single object ?, and homset G(?, ?) = G with F (?) = X. This immediately motivates one to generalize this to a functor F : C → D. This is precisely what is meant by an action of a category as defined and utilized in this work. The analogy immediately suggests an abstract Definition 98.
Definition 98 (Abstract or arrow theoretic) A transformation representation of a category C on X is a functor from C into a category D where X = qX∈Ob(C) F (X) is the object of objects (or a coproduct of objects) in D. c 2018, Indian Institute of Technology Delhi
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It is assumed that such a coproduct of the codomain category D exists which implies that such a representation induces an action on the multiple objects of F (C). Such an abstract (left and right) action gives rise to the base structured categories R ¯ as studied in Chapter 3. The abstract definition can be made X oF C and C F concrete or set-theoretic by considering small category and its action on the objects as structured sets and morphisms as structure preserving functions. Thus concrete R ¯ (left and right) action gives rise to the base structured categories X oF C and F C
as as studied in Chapter 3
5.1.1
Sets as categories
In category theory sets are often viewed as categories. More precisely a category is a set if all morphisms are identity morphisms and referred to as discrete category. This is possible using a standard inclusion ι : Set → Cat. Thus every contravariant functor thought of as a covariant functor (or a presheaf) F : Cop → Set can be completed to a fibred category which is said to be fibred in sets. Thus the base structured categories which arise through construction or completion of functors such F¯ U F¯ U ¯ 0 : Cop − ¯ : Cop − → D − → → D − → Set are truly nothing but completion of F as F R ι ¯ where ιF ¯ = F ¯ 0 . However we will suppress the notation Set → − Cat which is C ιF ι unless explicitly required within a context since Grothendieck completion of Setvalued functors is well studied (via the understood inclusion); refer [BW90]. The set-theoretic groupoid based definitions are taken from [Goe09] for the general category action.
Definition 99 [Goe09] Let C be a set and C (2) ⊆ C × C. Then C is a small category if there are maps (g, f ) 7→ gf from C (2) into C such that: • associativity: If (h, g) and (g, f ) belong to C (2) then (hg, f ) and (h, gf ), also belong to C (2) and we have (hg)f = h(gf ). • identity: the set of elements with (f, f ) = f denoted as C (0) • unit laws: If f ∈ C then (r(f ), f ), (f, s(f )) ∈ C (2) , r(f )f = f , and f s(f ) = f . c 2018, Indian Institute of Technology Delhi
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where the map r : C → C (0) such that r(f ) = f f −1 is called the range map and the map s : C → C (0) such that s(f ) = f −1 f is called the source map. when (g, f ) ∈ C (2) then g and f are called composable, or the set C (2) is the set of composable pairs. In the case when we have an additional map f 7→ f −1 from C into C such that for every f ∈ C we have (f −1 )−1 = f , then such a category is called a groupoid and f −1 is the inverse of f . Note that this definition is equivalent to the one we have recalled in Chapter 2 with objects as elements belonging to C (0) while the morphisms are elements of C. Corresponding to the Definition 99 we can now define a category action on a set. Definition 100 [Goe09] Suppose C is a small category and X is a set. Then a category C action (left) on X, is a surjection rX : X → C (0) and a map (g, x) 7→ g · x from C ∗ X := {(f, x) ∈ C × X : s(f ) = rX (x)} to X such that 1. if (f, x) ∈ C ∗ X and (g, f ) ∈ C (2) , then (gf, x), (g, f · x) ∈ C ∗ X and g · (f · x) = gf · x, 2. and rX (x) · x = x for all x ∈ X. In the same manner for the right action we use sX to denote the map from X to C (0) and then define the action on the set X ∗ C := {(x, f ) ∈ X × C : sX (x) = r(f )}. However going further we will not utilize this form of a category definition and the action since we feel it obscures the underlying arrow approach of the whole category theory by treating it as a set with some special set of axioms.
5.2
Base structured categories in symmetry
By relating the contemporary action or transformation groupoid to base structured category we gain an added perspective in the context of symmetries especially regarding the differences between local and global symmetry in basic examples and c 2018, Indian Institute of Technology Delhi
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also cases involving some sort of symmetry hierarchy. With Proposition 71 at our disposal we can now illustrate the well-known difference in the modeling of local and global symmetries through choice of different base categories.
5.2.1
Global symmetry using a single object base category
A homogeneous structure of any sort could be modeled as an object of a general category D (abstract or concrete). Then a single object base category and the corresponding base structured categories serve to capture the essence of global structure or F I symmetry on the homogeneous object. More precisely, let F : C − →D→ − Cat be the functor with C being a single object category with isomorphisms or more precisely an abstract group (or equivalently a pointed and connected groupoid). In this case, we have the image category F (C) containing a single object X which is precisely the homogeneous structure whose symmetry we are modeling. In the terminology of fibration there is a single fibre E? within the total category. This necessarily implies that the cartesian lifts of a base arrow are nothing but restrictions of morphisms such as (f, F f ) on the whole underlying set X when D is Set or any concrete category. Let us illustrate the idea so far using an example of a cyclic group of order 3, G3 acting on a simple finite set X = {1, 2, 3} using visual techniques of [Car09]. Here D F I is chosen as FinSet. We have F : G3 − → Finset → − Cat and we consider abstract action as defined in Chapter 3 of thesis. Note that this abstract action is practically F U expressed as a concrete action F : G3 − → Finset − → Set using the standard concrete base-structured category X oF G3 again defined in Chapter 3 of thesis which is nothing but essentially same as the classic transformation groupoid as we have seen earlier. However we will form base structured category of only abstract action which implies not decomposing the object X into (underlying) elements to emphasize the fact that it is considered as one single global object or precisely single basepoint (say ?) corresponding to a global homogeneous structure X . In this illustration X = {1, 2, 3} while the category X oF G3 consists of: • objects: a single object (?, X) denoted by Ob(X oF G3 ) R • morphisms: the collection G3 F ((?, X), (?, X)) = {(r, idX ) : (?, X) → (?, X)} c 2018, Indian Institute of Technology Delhi
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{1, 2, 3}
r¯
/
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r¯
2/
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r¯2
X id
%
r¯(X)
/X
r¯(X)
/3 X
r¯2 (X)
?
r
/?
r
/3 ?
r2
Figure 5.1: symmetry of a set using group as a base category. • identity: for the single (?, X), the morphism id(?,X) = (id? , idX ) • composition: if (r, r2 ) 7→ r ◦ r2 in G3 then ((r, idX ), (r2 , idX )) 7→ (r, idX ) ◦ (r2 , idX ) = (r ◦ r2 , idX · idX ) • unit laws: for (r, idX ), (id? , idX ) • (r, idX ) = (r, idX ) = (r, idX ) • (id? , idX ) • associativity: (r2 , idX ) • ((r, idX ) • (r, idX )) = ((r2 , idX ) • (r, idX )) • (r, idX ) = (r2 , idX ) • (r, idX ) • (r, idX ) We wish to remind the reader that although this category appears to have trivial arrows in its second component there is truly underlying action of the functors such as Fr on the category F(?) = X as studied in Chapter 3 of thesis. In case of confusion, one might choose to view this category as (F, C, D). Thus a single object multi-arrow base category introduces a specific (global) symmetry structure on X as shown in Figure 5.1 expressed by X oF G3 which intuitively signifies the fact that the structure of X is expressed relative to a chosen base.
5.2.2
Local and global symmetries using a multi-object base category
A non-homogeneous structure of any sort could be modeled as a collection of homogeneous components or multiple objects of a general category D (abstract or concrete). Then a multiple objects base category and the corresponding base structured categories serve to capture the essence of local and global structures or symmetries on the c 2018, Indian Institute of Technology Delhi
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{1}
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/
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r¯
3/
{3}
r¯2
x1 id?1
&
r¯(x2 )
/ x2
r¯(x3 )
r¯2 (x3 )
?1
r
x / ?2 id?2
r
/ 3 x3 x /3 ?3 id?3
r2
Figure 5.2: Symmetry of a set using groupoid as a base category. F
I
complete object. More precisely, let F : C − →D→ − Cat be the functor with C being a multi-object category with isomorphisms or more precisely a non-pointed arbitrarily connected groupoid. In this case, we have the image category F (C) containing multiple objects X1 , X2 , X3 ... or X = X1 q X2 q X3 q ... which is precisely the complete non-homogeneous structure whose symmetry we are modeling. In the terminology of fibration there are multiple fibres of the type E?1 in the total base structured category. This implies that cartesian lifts of a base arrow are local transformations only defined on these local objects or spaces. Let us again illustrate this idea using an example of a groupoid G3 acting on a F I simple finite set X = {1, 2, 3}. We have F : G3 − → Finset → − Cat and we consider abstract action again. Here the emphasis is on the fact that we decompose the object X into (underlying) elements or X = x1 q x2 q x3 . Thus same symmetry structure on X can be expressed by changing the base-structured category to a groupoid with multiple objects where the morphisms are same as those of a group. Hence X automatically gets partitioned by this base providing an illustration of how same symmetry is expressed by choosing a multi object base. In essence we have multiple basepoints (say ?1 ,?2 ,...) corresponding to a disjoint union of locally homogeneous structures X = x1 q x2 q .... Here B = G3 , a groupoid with three distinct objects as shown in the base while the category X oF G3 is defined as: • objects (?1 , x1 ), (?2 , x2 ), (?3 , x3 ) denoted by Ob(X oF G3 ) • morphisms the collection X oF G3 ((?1 , x1 ), (?2 , x2 )) = {(r, idx2 ) : (?1 , x1 ) → c 2018, Indian Institute of Technology Delhi
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(?2 , x2 )} • identity: for each object (?1 , x1 ), the morphism id(?1 ,x1 ) = (id?1 , idx1 ) • composition: (r, idx3 ) • (r, idx2 ) = (r ◦ r, idx3 · Fr(idx2 )) = (r2 , idx3 ) • unit laws: for (r, idx2 ), (id?2 , idx2 ) • (r, idx2 ) = (r, idx2 ) = (r, idx2 ) • (id?1 , idx1 ) • associativity: (r2 , idx1 ) • ((r, idx3 ) • (r, idx2 )) = ((r2 , idx1 ) • (r, idx3 )) • (r, idx2 ) = (r2 , idx1 ) • (r, idx3 ) • (r, idx2 ) Thus a multi-object connected base now models same symmetry but from within the object since X is no longer treated as single relative to the base but as co-product or disjoint union of three distinct singleton objects X = x1 q x2 q x3 as shown in Figure 5.2. Hence given set X when treated as a single object requires a group (or equivalently a single object category with invertible arrows) to express its symmetry or structure. The same set X when treated as multi-object (commonly termed as object of objects) i.e X = x1 q x2 q x3 requires a multi-object groupoid as a base category to express the same symmetry.
5.3
Hierarchy of symmetry I: Leyton’s generative theory
In this section we describe the difference in the modeling of generative theory of shape proposed in [Ley86a],[Ley86b],[Ley86c],[Ley01] using groups, groupoids and base structured categories. The model using hierarchy of groups leading to wreath groups is utilized by Leyton in proposing the generative theory of shape; see [Ley01] and references therein. However the same theory could be generally modeled using hierarchy of groupoids following [Wei96]. Consider a simple example of a non-homogeneous structure X = X1 ∪ X2 as shown in Figure 5.3 where we have disjoint union of two homogeneous structures X1 and X2 . The complete symmetry of such a structure is well studied in the literature [Wei96] and captured by groupoid. What we wish to illustrate in this section is that c 2018, Indian Institute of Technology Delhi
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R2
x1 x3
x4 O
x2
x6
x5
Figure 5.3: Non homogeneous Structure X = X1 ∪ X2 illustrating 2-level symmetry. R2 x1 O1 x3
x2
Figure 5.4: Symmetry of prototype X1 captured by D3 . the symmetry of such a structure was shown to be also captured by a wreath group in [Ley01]. We first study how this modeled whenever possibile and then make a comparative analysis with the standard groupoid model to better understand the difference utilizing the perspective offered by the base structured categories. In [Ley01] the whole non-homogeneous structure X (generated by an artist) is considered to be the transfer of homogeneous prototypes such as X1 in the example we have considered. The homogeneous prototypes are modeled independently in their own separate global spaces. Thus in Figure 5.4 the symmetry of prototype X1 is captured in Euclidean space R2 by the familiar group D3 (which truly acts on the whole space R2 or in other words a subgroup of O(2)).
c 2018, Indian Institute of Technology Delhi
5.3 Hierarchy of symmetry I: Leyton’s generative theory
injection
X
universal
X1 ∪ X2
pullback couniversal
149
X × Ob(Z2 )
Permutation action D3
CoProduct action Extended action projection pullback w 2 D3 Z D3 × D3 couniversal couniversal
Figure 5.5: Leyton’s generative model using groups Hence two local symmetries are captured by dihedral groups of order three. The first local symmetry is captured by the permutation action of group D3 on X1 where X1 = {1, 2, 3} = {x1 , x2 , x3 }. Similarly the second local symmetry is captured by the permutation action of another group D3 on X2 where X1 = {4, 5, 6} = {x4 , x5 , x6 }. F I Thus we have two functors which denote actions of type F : D3 − → Set → − Cat. Now since a group when considered as one object category (refer Section 5.7 for single object species of structures) always determines a symmetry of global nature, the morphisms determined by its elements are well defined on every global point of the object X. For a collection of multiple independent objects X = X1 ∪ X2 with individual symmetries, the global symmetry of this collection becomes the product of individual symmetries, since the collection gets treated as a single object X with no specific partition or injections of subobjects giving rise to a product category F I D3 × D3 = B2 or a single functor F2 : D3 − → Set → − Cat or as a base category X oF2 (D3 × D3 ). Here since all the objects fuse into a single object (correspondingly captured by single object of base category B2 ), there remains a single base point relative perspective. Finally the global symmetry is captured by the action of a reflection group (reflection across y-axis of Euclidean plane) of order two which we will set-wise as Z2 = {e, t}. The notation X1 ∪ X2 = X1 × Ob(Z2 ) signifies the fact that complete object is a Leyton transfer of the 3 element set X1 as a prototype. To summarize, here on a single object X = {1, 2, 3, 4, 5, 6} we have first action of D3 × D3 or the coproduct action in the group theory, which captures the local structure or symmetries but from a global perspective where the given set Y is considered as one object ?. The next level of structure or symmetry between the local symmetries is captured by action of Z2 on D3 ×D3 by automorphisms, leading to a wreath group. The 2-hierarchy of symmetries is therefore captured by a group viewed as a single c 2018, Indian Institute of Technology Delhi
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w C2 which is a special case of semi-direct group (C3 ×C3 )oC2 . object category C3 This is the restriction to category of groups or global symmetries. Now we capture the symmetry of the same non-homogeneous structure using the widely known and studied groupoid model following [Wei96]. In this model the whole object X is modeled as contained in the single space R2 . As we have seen that every group action results into a transformation groupoid let us denote the groupoid formed by the action of group Γ of rigid motions of R2 on R2 as G(Γ, R2 ) = {((x, y), γ)|x ∈ R2 , y ∈ R2 , γ ∈ Γ, x = γy}. Then the groupoid leaving the X invariant is just G(Γ, R2 )|X consisting of those g ∈ G(Γ, R2 ) where the domain and codomains of the g are simply restricted to X. This is the groupoid capturing the reflection symmetry and is isomorphic to the action groupoid produced from action of reflection group Z2 = {e, t} on the set X. The local symmetry is captured by another groupoid Gloc . To define this groupoid we consider the plane R2 as the disjoint union of X1 = X ∩ X1 , X2 = X ∩ X2 and R2 \ X (plane puncturing out the six points). More precisely Gloc = {((x, y), γ)|x ∈ X, y ∈ X, γ ∈ Γ, x = γy} for which y has a neighborhood u in R2 such that γ(u ∩ Xi ) ⊆ Xi . This groupoid captures the dihedral local symmetries of X1 and X2 . To connect with the generative theory of Leyton; the local and global groupoids could be viewed as base (groupoid) structured categories as shown in Figure 5.2. A groupoid being a multiple object category (refer Section 5.7 for multi-object species of structures) captures a symmetry of a local nature since the morphisms determined by its elements are locally defined on specific local point of the object X. Thus a collection of multiple independent objects X = X1 ∪ X2 with individual symmetries, the global symmetry of this collection becomes the amalgamation of individual local symmetries. The collection gets naturally treated as a collection of multiple objects X with partition or injections of subobjects X1 , X2 giving rise to a coproduct category B2 which is the Coproduct Groupoid D3 q D3 . Intuitively the separation of objects {1, 2, 3} and {4, 5, 6} (correspondingly captured by multiple objects of base category B2 ) keeps multi-basepoints relative perspective intact in this structure allowing the possibility of adding arrows across different
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objects locally. Indeed every individual object in the whole co-product object retains its individuality and the domain and range set remains divided into multiple subsets on which arrows get defined. In summary, here the given set X is no longer treated as a single object but a hierarchy of multi-objects leading to global, local, sub-local perspective on symmetries of X (the example we are considering, will result into two levels of Base corresponding to two hierarchy levels of symmetries or structures). Roughly speaking this is akin to horizontal categorification where we break a single global object X into multiple sub-objects X1 and X2 viewed as fibred on B1 = Z2 with two objects ∗1 and ∗2 . The fibres containing single objects X1 and X2 respectively are further broken into second level of sub-objects {1},{2},{3} and {4},{5},{6} viewed as fibred on D3 and D3 with three objects in each of the base groupoids. The notation X1 ∪ X2 = X × Ob(Z2 ) signifies the fact that complete object is an Ehresmann transport (see [Bro87] and [Ehr84]) of the 3 element set X1 as a prototype. injection
X
universal
X1 ∪ X2
pullback couniversal
Permutation action CoProduct action injection (D3 q D3 ) = B2 D3 universal
X × Ob(Z2 )
Extended action pullback B2 o Z2 couniversal
Figure 5.6: Leyton’s generative model using groupoids Noting that actions of categories give rise to base structured categories, we get an equivalent model as shown in Figure 5.7 using base structured categories. Thus, X o D3
injection universal
X1 o D3 q X2 o D3
pullback couniversal
(X1 o D3 q X2 o D3 ) o Z2
Figure 5.7: Leyton’s generative model using base structured categories natural generativity and transfer could be more faithfully modeled replacing category theoretic analogues of the sets and groups giving rise to base structured categories as shown in Figure 5.8 which makes use of the perspective of functor as a category action. c 2018, Indian Institute of Technology Delhi
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(Y,FY) as fiber
(X,FX) as fiber
Base category C (as control)
Figure 5.8: Base structured categorical X oF C model of Leyton’s generative theory
In [HW99] the authors have raised concerns on the idea of successive referencing. It appears to us at this moment that the issue might be tackled using the notion of base structured categories along with hierarchy of bases leading to n-categories; refer Section 5.4 and Section 5.5 for the details.
5.4
Hierarchy of symmetry II: Base structured categorical models
In continuation of last section, here we revisit the modeling of arbitrary multiple levels of symmetry (widely known as local/global or internal/external) algebraically using groups, groupoids and base structured categories.
5.4.1
Hierarchy of symmetries
For considering the hierarchy of bases we utilize notations with subscripts 1, 2, ....n to denote a level in hierarchy. Thus B1 will denote a base category at the first (outermost) level whereas X1 denotes coproduct object at first level. The subscripts c 2018, Indian Institute of Technology Delhi
5.4 Hierarchy of symmetry II: Base structured categorical models
{1, 2, 3, 4, 5, 6}
r×r(Y )
/
{3, 1, 2, 6, 4, 5}
r×r(Y )
/
153
1 {2, 3, 1, 5, 6, 4}
r2 ×r2 (Y )
Y id?
%
r×r(Y )
/Y
r×r(Y )
/2 Y
r2 ×r2 (Y )
?
r×r
/?
r×r
/2 ?
r2 ×r2
Figure 5.9: Inner symmetry using groups a, b, ... will be used to denote the objects or base categories at a given level. Thus X1a and X1b will denote two objects within the coproduct object X1 whereas B2a , B2b denotes base categories at second level (inner) level. Hence a 2-level fibration of objects will be denoted by [(X2a oF2a B2a ) q (X2b oF2b B2b ) q ...] oF1 B1 where X1a = (X2a oF2a B2a ), X1b = (X2b oF2b B2b ), (X1a q X1b q ...) = X1 and B2 = (B2a q B2b q ...). Again taking the same example of Section 5.3 where we have independent copies of X say X1 and X2 which are translates of each other forming a total set Y = X1 q X2 , the complete symmetry or structure of Y can be expressed or formulated again using groups, groupoids and corresponding base structured categories.
5.4.2
Using hierarchy of Groups [(X oF2 (G3 × G3 )] oF1 G2
In this model second level (inner) base B2 = G3 × G3 , the direct product group is a F2 I single object category with F2 : B2 −→ FinSet → − Cat with F2 (?) = Y = X1 ∪ X2 = {1, 2, 3, 4, 5, 6} and Y oF2 G3 × G3 consisting of: • objects: a single object (?, Y ) denoted by Ob(Y oF2 G3 × G3 ) • morphisms: the collection (Y oF2 G3 × G3 )((?, Y ), (?, Y )) = {(r × r, idY ) : (?, Y ) → (?, Y )} • identity: for the single (?, Y ), the morphism id(?,Y ) = (id? , idY ) • composition: if (r × r, r2 × r2 ) 7→ (r × r) ◦ (r2 × r2 ) in G3 × G3 then ((r × r, idY ), (r2 × r2 , idY )) 7→ (r × r, idY ) ◦ (r2 × r2 , idY ) = ((r × r) ◦ (r2 × r2 ), idY · idY ) c 2018, Indian Institute of Technology Delhi
5.4 Hierarchy of symmetry II: Base structured categorical models
(?, Y )
t¯(?,Y )
(r×r,idY )
(r2 ×r2 ,idY )(?, Y
)
%
∗
t¯(?,Y )
/
(?, Y )(r2 ×r2 ,idY )
(r×r,idY )
id∗
(?, Y )
(r×r,idY )
(r×r,idY )
(?, Y )
/
154
t¯(?,Y )
t
/
(?, Y ) /
∗
Figure 5.10: Outer symmetry using groups • unit laws: for (r × r, idY ), (id? , idY ) ◦ (r × r, idY ) = (r × r, idY ) = (r × r, idY ) ◦ (id? , idY ) • associativity: (r2 × r2 , idY ) ◦ ((r × r, idY ) ◦ (r × r, idY )) = ((r2 × r2 , idY ) ◦ (r × r, idY )) ◦ (r × r, idY ) = (r2 × r2 , idY ) ◦ (r × r, idY ) ◦ (r × r, idY )
In this model first level (outer) base B1 = G2 , the cyclic group of order two, is a F1 I single object category and F1 : G2 −→ Cat → − Cat and F1 (∗) = Y oF2 (G3 × G3 ) with [(Y oF2 (G3 × G3 )] oF1 G2 consisting of: • objects: a single object (∗, F1 (∗)) = (∗, ?, Y ) denoted by Ob([(Y oF2 (G3 × G3 )] oF1 G2 ) • morphisms: the collection of form ((∗, ?, Y ), (∗, ?, Y )) = {(t, (r × r, idY )) : (∗, ?, Y ) → (∗, ?, Y )} • identity: for the single (∗, ?, Y ), the morphism id(∗,?,Y ) = (id∗ , (id? , idY )) • composition: if (t, t) 7→ t • t in G2 then ((t, (r × r, idY )), (t, (r × r, idY ))) 7→ (t, (r × r, idY )) • (t, (r × r, idY )) = (t • t, ((r × r) ◦ (r × r), idY ))
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5.4 Hierarchy of symmetry II: Base structured categorical models
{1}
r¯
/
{2}
r¯
3/
{3}
{4}
r¯
r¯2
x1 id?1
&
r¯(x2 )
r¯(x3 )
/ x2 x
r
{5}
r¯
3/
{6}
r¯2
r¯2 (x3 )
?1
/
155
/ ?2 id?2
/ 3 x3 x
x4
r
/3 ?3 id?3
id?4
&
r¯(x5 )
/ x5
r¯(x6 )
r¯2 (x6 )
?4
r
r2
x / ?5 id?5
r
/ 3 x6 x /3 ?6 id?6
r2
Figure 5.11: Inner symmetries using groupoids The usual unit laws and associativity axiom can be verified easily. Henceforth we will not mention the unit laws and associativity unless explicitly required within a particular context.
5.4.3
Using hierarchy of groupoids [(X2a oF2a G3 ) q (X2b oF2b G3 )] oF1 G2
In this model second level (inner) base B2 = G3 q G3 , a groupoid with six distinct objects as shown in the base of Figure 5.11. In this model F2 : (G3 q G3 ) → FinSet which is same as F2 = F2a q F2b and Y = X1 ∪ X2 . Then X2 oF2 B2 is a coproduct of two base structured categories which is (X2a oF2a G3 ) q (X2b oF2b G3 ) consisting of: • objects (?1 , x1 ), (?2 , x2 ), (?3 , x3 ), (?4 , x4 ), (?5 , x5 ), (?6 , x6 ) • morphisms the collection ((?1 , x1 ), (?2 , x2 )) = {(r, idx2 ) : (?1 , x1 ) → (?2 , x2 )} Note that there will not be any arrows across indices’s (1,2,3) to (4,5,6) since the two base groupoids are not connected. • identity: for each object (?1 , x1 ), the morphism id(?1 ,x1 ) = (id?1 , idx1 ) • composition: (r, idx3 ) • (r, idx2 ) = (r ◦ r, idx3 · F2 r((idx2 )) = (r2 , idx3 ).
The usual unit laws and associativity axiom are straightforward.
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5.4 Hierarchy of symmetry II: Base structured categorical models
(?1 , x1 )
F1 t(?1 ,x1 )
(r,idx2 )
(r2 ,idx3 )
(?2 , x2 )
F1 t(?2 ,x2 )
&
∗1
/
(?5 , x5 )
(r2 ,idx6 )
(r,idx6 )
id∗1
(?4 , x4 )
(r,idx5 )
(r,idx3 )
(?3 , x3 )
/
156
F1 t(?3 ,x3 )
t
/
(?6 , x6 ) x / ∗2 id∗2
Figure 5.12: Outer symmetry using groupoids In this case the first level (outer) base B1 = G2 , the cyclic groupoid of order F1 I two, is a two object category with two arrows and F1 : G2 −→ Cat → − Cat and F1 (∗1 ) = (X2a oF2a G3 ) while [(X2 oF2 (G3 q G3 )] oF1 G2 consists of: • objects: the pairs of type (∗1 , (?1 , x1 )) denoted by Ob([(X2 oF2 (G3 qG3 )]oF1 G2 ) • morphisms: [(X2 oF2 (G3 q G3 )] oF1 G2 ((∗1 , (?1 , x1 )), (∗2 , (?5 , x5 ))) are pairs {(t, (r, idx4 )) : (∗, (?, Y )) → (∗, (?, Y ))} where t : ∗1 → ∗2 ∈ G2 , (r, idx4 ) : F1 t(?1 , x1 ) → (?5 , x5 ) • identity: for (∗1 , (?1 , x1 )), the morphism id(∗1 ,?1 ,x1 ) = (id∗1 , (id?1 , idx1 )) • composition: (t, (r, idx3 )•(t, (r, idx5 ) = (t◦t, (r, idx3 )·F1 t(r, idx5 )) = (id∗1 , (r2 , idx3 )) since F1 t(r, idx5 ) = (r, idx2 )
The simple example of permutation of finite sets illustrated that hierarchy of algebraic structures could be modeled using hierarchy of appropriate base structured categories. It follows naturally that base structured categories from groupoid bases over group bases in modeling hierarchy of symmetries are preferred due to the following heuristic reasons:
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• The ability to incorporate local non-uniform symmetries using multi-object and arbitrarily connected groupoid base. • Ease of extraction both in terms of computational efficiency, parallel processing and non-redundant processing in a groupoid base. • The ability to proceed in distinct steps by separation of objects at each step is naturally provided by the hierarchy of groupoids rather than groups.
5.4.4
Connection of hierarchy of base structured categories with standard composite of Grothendieck fibrations
The hierarchy of base structured categories are naturally connected with the classic composite fibration since multi-level base structured category is a special case of composite fibration of categories. First we recall the classic composite of fibration Lemma from [Jac99]. Lemma 101 [Jac99] Let P : E → B2f ib and Q : B2f ib → B1f ib be the usual fibrations; then 1. Composite functor QP : E → B1f ib is a usual proper fibration where f in E is QP -cartesian ; f is P -cartesian and P f is Q-cartesian. Further P and Q both being cloven or split, the composite fibration QP is respectively cloven or split; and 2. For every object I ∈ B1f ib , we obtain by restriction a functor PI : EI → B2I f ib where EI = (QP )−1 (I) and B2I f ib = Q−1 (I). All such restriction functors PI s are themselves fibrations. Proof : Refer Lemma 1.5.5 of [Jac99]. Observe that B1f ib of the classic case is same as B1 in the hierarchy of base structured categories. Now since in hierarchical base structured category; each object is itself a base structured category which is also naturally an object of Cat we have standard split fibration with X1a , X1b as objects of Cat. Thus if we denote the pseudofunctor corresponding to split fibration QP by Ψ : B1 op f ib → Cat, then Ψ(I) = c 2018, Indian Institute of Technology Delhi
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X1a and so on. Now since each base structured category X1a also has a corresponding base category B2a , we naturally also have a functor F1B : B1 → Cat with F1B (I) = B2a where I is the object of B1 in the usual base category of classic fibration. Hence the total category B2f ib of the classic split fibration Q is given by B2f ib = B2 oF1 B B1 where B2 = (B2a q B2b q ...) is the coproduct of second level base categories. Thus E = [(X2a oF2a B2a ) q (X2b oF2b B2b ) q ...] oF1 B1 .
5.5
Hierarchy of structures: 2-groups to n-category theory
In this section we utilize the results of [JCM14] especially that a 2-group captures the symmetry of a category and results into a transformation double category. We show a hierarchy of 2-groups capturing the symmetry of hierarchy of base structured categories as studied in Section 5.4 resulting into hierarchy of transformation double categories such as (X2a oF2a B2a )//2G, (X2b oF2b B2b )//2G within (X1 oF1 B1 )//2G naturally leading us into n-groups or higher category theory. Our primary motivation for exploring such a connection is to get some insights on structure of possible algorithms in connection with signal representation since it seems to us at this moment that such a multi-level structure is akin to multi-resolution analysis in wavelet signal representation. The precise connections will become clear only in the future. We start by recalling the definition of a 2-group. Definition 102 [JCM14] A 2-group 2G is a 2-category with a single object (2G(0) = {?}), for which all 1-morphisms γ ∈ 2G(1) and 2-morphisms χ ∈ 2G(2) are invertible.
This definition is studied in [JCM14] through connection with an equivalent categorical group and crossed module. Refer to [JCM14] and references therein for a review of symmetry using category theory or the categorification of local and global symmetry. We demonstrate the transformation double categories produced at each level by a hierarchy of 2-groups which correspondingly capture the symmetries of c 2018, Indian Institute of Technology Delhi
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base structured categories at each level. This leads one into the realm of n-groups and higher categories. The outermost 2-group is denoted as in Equation 5.1 where γo and γo0 are its 1-morphisms whereas χo is the 2-morphism. Similarly the inner 2-groups for the inner base structured categories are shown in Equation 5.2. γo
y
(X1 oF1 B1 ) e
χo
(X1 oF1 B1 )
(5.1)
γo0
γ1
x
x
(X2a oF2a B2a ) f
γ2
χ1
γ10
(X2a oF2a B2a ) (X2b oF2b B2b ) f
χ2
(X2b oF2b B2b )
γ20
(5.2) Thus hierarchy of base structured categories naturally leads one into the domain of higher category theory (see both [Lei03] and [BM10] and references therein) where each level of symmetry or base structured category has an automorphism group given by a 2-group (or equivalently a categorical group or a crossed module) all in a hierarchical fashion. This leads to an interesting structure of a hierarchy of naturality squares within naturality squares as shown in Figure 5.13. Thus a 2-level hierarchy results into 3-morphisms (γo , γo0 ) and 4-morphisms (χo ) of outer 2-group relative to 1-morphisms (γ1 , γ10 ,) and 2-morphisms (χ1 ) of inner 2-groups. We now use the example of [(X1 oF2a G3 ) q (X2 oF2b G3 )] oF1 G2 to show how the fundamental cells or squares corresponding to (χ, f ) ∈ 2G × C from [JCM14] appear
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5.5 Hierarchy of structures: 2-groups to n-category theory
2 1
2
2 4
1
1
3
1
160
2
1
3
2
4
1
1
2
2
1
2
Figure 5.13: Hierarchy of naturality squares leading to 3-,4-morphisms in a hierarchy starting with the inner 2G to outer 2G.
2G
Φo
/ φˆo
Aut(X oF1 G2 ) &
(5.3)
X oF1 G2 Here φˆo : 2G × (X oF1 G2 ) → (X oF1 G2 ) is the 2-group action. Now Φo is a 2functor from 2G to 2-category of invertible endofunctors on category (X oF1 G2 ). The fundamental unit (or cell or naturality) square of co-domain endofunctor category is the square corresponding to a morphism (χ, f ) ∈ 2G × C which is the action of 2G on C; see [JCM14].
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5.5.1
161
The naturality square corresponding to inner 2-group action
First the symmetry of X1 oF2a G3 is expressed using an (inner) 2-group as shown in Equation 5.4.
Φ
2G
/
Aut(X1 oF2a G3 )
φˆ
'
(5.4)
X1 oF2a G3 The 2-functor Φ defines a mapping on the single object given by Φ(?) = (X1 oF2a G3 ). The mapping of 1-morphism is denoted as Φ(γ) while that of the 2-morphism is denoted as Φ(γ, χ) where χ : γ → γ 0 is the 2-morphism of the 2G.
Φ(γ)=γI()
z
X oF2a d G3
Φ(γ,χ)
X oF2a G3
(5.5)
Φ(γ 0 )=γ 0 I()
The action of 1-morphisms γ, γ 0 and 2-morphism χ between these on a given arrow
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(r, idx2 ) in X1 oF2a G3 generates a fundamental square as shown in Equation 5.6.
(r,idx2 )
(?1 , x1 )
/
(?1 , x1 ))
(?2 , x2 )
(γ,(?2 ,x2 ))
(χ,(r,idx2 ))
(γ 0 ,(?1 ,x1 ))
γI(r,idx2 )
γ I (?1 , x1 )
-/
(r,idx2 )
(γ,(?1 ,x1 ))
(?2 , x2 )
/
(γ 0 ,(?2 ,x2 ))
γ I (?2 , x2 ) Φ(γ,χ) ((?2 ,x2 ))
Φ(γ,χ) ((?1 ,x1 )) 0
(
γ I (?1 , x1 )
γ 0 I(r,idx2 )
- 0( / γ I (?2 , x2 )
(5.6)
5.5.2
The naturality square at a node of outer naturality square
Next at each node of outer naturality square as shown in Figure 5.13 we have inner naturality squares corresponding to inner 2-group actions. Thus the symmetry of (∗1 , X1 oF2a G3 ) could be expressed using an induced (in-out) 2-group structure as shown in Equation 5.7.
(id∗1 ,γ)
x
(∗1 , X oF2a G3 ) f
(id∗1 ,χ)(∗1 , X
oF2a G3 )
(5.7)
(id∗1 ,γ 0 )
Corresponding to an inner natural square as shown in Equation 5.6 we get a natural square at the node corresponding to object ∗1 of the outer groupoid G2 as c 2018, Indian Institute of Technology Delhi
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shown in Equation 5.8.
(id∗1 ,(r,idx2 ))
(∗1 , (?1 , x1 ))
/ (∗1 , (?2 , x2 ))
(∗1 , (?1 , x1 ))
(id∗1 ,(r,idx2 ))
(id∗1 ,(γ,(?1 ,x1 )))
(id∗1 ,(γI(r,idx2 )))
(∗1 , (γ I (?1 , x1 )))
(∗1 , (?2 , x2 ))
(id∗1 ,(γ,(?2 ,x2 )))
(∗1 ,(χ,(r,idx2 )))
(γ 0 ,(?1 ,x1 ))
-/
/
(γ 0 ,(?2 ,x2 ))
(∗1 , (γ I (?2 , x2 ))) (id∗1 ,(Φ(γ,χ) ((?2 ,x2 ))))
(id∗1 ,(Φ(γ,χ) ((?1 ,x1 ))))
*
-
(∗1 , (γ 0 I (?1 , x1 )))
(id∗1 ,(γ 0 I(r,idx2 )))
/
*
(∗1 , (γ 0 I (?2 , x2 )))
(5.8)
5.5.3
The naturality square corresponding to outer 2G action
Finally the whole symmetry of X oF1 G2 is expressed using an (global) 2-group (which truly becomes a 4-group relative to inner 2-groups) as shown in Equation 5.3. The 2-functor Φ defines a mapping on the single object given by Φo (?) = (X oF1 G2 ). The mapping of 1-morphism is denoted as Φo (γo ) while that of the 2-morphism is denoted as Φo (γo , χo ) where χo : γo → γo0 is the 2-morphism of the outer (global) 2G. Φ(γo )=γo I()
{
X oF1 d G2
Φ(γo ,χoX )
oF1 G2
(5.9)
Φ(γo0 )=γo0 I()
The action of 1-morphisms γo , γo0 and 2-morphism χo between these on a given
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arrow (r, idx2 ) in X oF1 G2 generates a fundamental square as shown in Equation 5.10.
(t,(r,idx4 ))
(∗1 , (?1 , x1 ))
/
(∗1 , (?1 , x1 )) (γo ,(∗1 ,(?1 ,x1 )))
γo I(t,(r,idx4 ))
γo I (∗1 , (?1 , x1 ))
-/
(t,(r,idx4 ))
(∗2 , (?5 , x5 ))
(γo ,(∗2 ,(?5 ,x5 )))
(χo ,(t,(r,idx4 )))
(γo0 ,(∗1 ,(?1 ,x1 )))
(∗2 , (?5 , x5 ))
/ γo
(γo0 ,(∗2 ,(?5 ,x5 )))
I (∗2 , (?5 , x5 )) Φ(γo ,χo ) ((∗2 ,(?5 ,x5 )))
Φ(γo ,χo ) ((?1 ,x1 ))
)
γo0 I (∗1 , (?1 , x1 ))
γo0 I(t,(r,idx4 ))
- 0 ) /γ I o
(∗2 , (?5 , x5 ))
(5.10) Thus a simple example of finite sets with 2-levels of symmetry served to illustrate a nesting of 2-groups leading to a 4-group which captures its essential structure. Detailed connection of such base hierarchy with the existing literature [Lei03] [BM10] needs to be explored further. Understanding the complete ramification of such a connection from an applied perspective is an interesting research direction.
5.6
Geometries as base structured categories X oF C
As studied comprehensively in [Sha97], Felix Klein unified many geometries utilizing the notion of a principal group. First we recall the classic definition of Klein geometry.
Definition 103 [Sha97] A Klein geometry is a pair (G, H) where G is a Lie group and H ⊂ G a closed subgroup such that G/H is connected. G is called the principal group of the geometry. The coset space X = G/H is the space of the geometry or Klein geometry.
Proposition 104 Every Klein Geometry is a base-structured category. c 2018, Indian Institute of Technology Delhi
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Proof : Consider Klein geometry which is a pair (G, H) as in Definition 103. To show that it is a base-structured category first consider the closed lie subgroup H ⊂ G. Since every group could be considered as a category with one object in which every morphism is an isomorphism; the lie group H is viewed as a category H with a single object ? with Hom(?, ?) = H and the composition map on homomorphisms is the multiplication in H. Note that H is equivalently a group object in the category Man∞ of smooth manifolds and therefore its homset Hom(?, ?) is an object of Man∞ F U with this added structure. Now we can define a functor F : H − → Man∞ − → Set. The Grothendieck completion of this functor gives the base structured category G = X oF H. The image of single object ? under functor F is the smooth connected manifold X = F (?) which is same as the discrete category F(?) which is the space of the geometry (which we shall call a global space to emphasize the crucial fact that it is the image of single object in the base category). The category G is simply isomorphic to the transformation groupoid corresponding to the action of lie group G on the smooth manifold X or precisely X//G as defined in Definition 70. Hence Klein geometry (G, H) is essentially a base structured category G = X oF H. In a spirit of generalization, following the original motivation of Charles Ehrsemann [Ehr84], [Ehr08] which is reviewed in [Pra07] where the term groupoid geometry is first coined as far as we are aware, we give a simple definition of groupoid geometry using the perspective offered by the base structured category X oF C. Proposition 104 motivates one naturally to consider a lie groupoid replacing the classic lie group H. Thus we have a new base lie groupoid structured geometry generalizing the classic Klein geometry. Definition 105 A groupoid (or base groupoid structured) geometry is a pair (G, B) where G is a Lie groupoid and B ⊂ G a closed subgroupoid such that G/B is globally arbitrarily connected (locally connected). G will be called the principal groupoid of the geometry. The multi-object coproduct space X = G/B is the usual space of the geometry or (similar to Klein geometry) ‘groupoid geometry’ itself. The motivation comes from generalizing the geometry of the function or signal space (that corresponds to generative base structured signal spaces) from Euclidean c 2018, Indian Institute of Technology Delhi
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geometry (that corresponds to Hilbert spaces) With this definition, it should even be possible to conceive of a geometry which contains inner hierarchy of geometries using the hierarchy of symmetries or base categories as developed in the earlier Section 5.4. We now note down a quick comparison of this geometry with the existing notion of Klein geometry in Table 5.1. This serves to intuitively illustrate the more general properties of this geometry with a multi-origin arbitrarily connected spaces, all attributed to the concept of multi-object groupoid action. Klein Geometry Lie Groups (G, H) Principal group G = X o H Homogeneous coset space X = G/H Single basepoint(origin) x0 ∈ X H is stabilizer subgroup of x0 in G
Groupoid Geometry Lie Groupoids (G, B) Principal groupoid G = X o B Coproduct Space X = X1 q X2 q ... = G/B Multi-basepoints (origins) x1 , x2 , .. ∈ X Local groups in B stabilize x1 , x2 , ... in G
Table 5.1: From Klein geometry to groupoid geometry
Proposition 106 A groupoid geometry is a base-structured category G = X oF B. Proof : Consider groupoid geometry which is a pair (G, B) as in the Definition 105. To show that it is a base-structured category first consider the closed lie subgroupoid B ⊂ G. Since every groupouid could be considered as a category with multiple objects in which every morphism is an isomorphism; the lie groupoid B is viewed as a category B with many objects ?1 , ?2 , ... Here B is equivalently a groupoid object in the category F U Man∞ of smooth manifolds. Now we can define a functor F : B − → Man∞ − → Set. The Grothendieck completion of this functor gives the base structured category G = X oF B. The image of multiple objects ?1 , ?2 , .. under functor F are multiple smooth connected manifolds X1 = F (?1 ), X2 = F (?2 ), ... which are same as the multiple discrete categories F(?1 ), F(?2 ), ... which is collective space of the geometry or more precisely X = q?∈Ob(B) F (?) = X1 q X2 q ... (which we shall call a collection of local c 2018, Indian Institute of Technology Delhi
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spaces to emphasize the crucial fact that it is the image of multiple objects in the base category). Hence groupoid geometry (G, B) is essentially a base structured category G = X oF B where G, B are set-theoretic notations for the small categories G, B. With limited exposure to the history of development category theory, it appears to us that the concept of groupoid geometry was first conceived by Charles Ehresmann [Ehr84] and explicitly proposed in [Pra07] by generalizing smooth (locally trivial) principal bundle (in Ehresmann’s sense) to non-locally trivial bundles. Referring [Ehr08], Charles Ehresmann who showed the equivalence between various notions such as classic presheaves (Set valued functors), discrete fibrations and (mathematical) species of structures. In this work we try to generalize this to D valued functor F by using a suitable F or F for objects general than sets in connection with our applied work which utilizes Hilbert spaces as objects rather than just sets. Again referring [Ehr08], Ehresmann reformulated the fundamentals of differential geometry specifically using concepts of fibred spaces and foliated manifolds and some sort of a species of local structures and associated a category (groupoid) to it. Starting from [Pra07], we have tried to conceptualize the geometry by interpreting the work of [Ehr84] from the perspective of category action studied in Chapter 3 of this thesis. In doing so we have proposed a simple definition for a geometry as a base structured category which arises from a general category action. As far as we are aware, within the literature there is no such Definition 105 of a groupoid geometry. An interested reader well-versed on the subject of differential geometry along with considerable exposure to the category Diff ; can verify and completely develop this concept of base groupoid structured geometry (with a hierarchy of geometries using the hierarchy of bases we developed here giving rise to geometries within geometries). The details of connections with existing notion of non-commutative geometries [Con95] will be worked out in the future. Finally the reader must note that in [Ley01] the author argues against the Klein geometry using wreath group as implied base in situations involving transfer. It seems appropriate to us that the groupoid structured geometry (with hierarchy of groupoid bases) as defined here models the original conception of a generative geometry [Ley01] using hierarchy of multi-object groupoids than singleobject groups in a functorial way.
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168
Multi-object species of structures: Heuristic discussion
In this section referring [Bro87], [Joy81] we motivate the concept of multi-object species of algebraic structures which have some sort of natural collection of many objects and possibly might be related to each other. As mentioned in [Bro87] this term seems to have originated during Bourbaki seminars. We understood such a species of structures by characterizing them as having a base object containing multiple objects or equivalently as a functor F : C → D from a base category with multiple objects to a general category generalizing the spirit and definition of species of structures as endofunctors in [Joy81]. For this first we revisit what can be loosely termed as a sort of object-arrow duality which is implicitly present in this thesis using the notion of base structured categories that characterize a functor or in other words object (category) that essentially captures the structure of arrow (functor). Recall that (F, C, D) characterizes a functor as a structure preserving morphism and captures the essential structure of F . As an illustration consider the special example of an identity functor idC . This functor can be treated as a category C which is just reflecting the fact that every identity arrow of an object can be taken to be the object itself. However the base structured category (idC , C, C) also serves to capture the structure of idC . Indeed by construction this is a category which consists of, • objects: (X, X), (Y, Y ), ... denoted by Ob(GidC ) • morphisms: collection GidC ((X, X), (Y, Y )) = {(f, f ) : (X, X) → (Y, Y )} • identity: for each (X, X), the morphism 1(X,X) = (1X , 1X ) • composition: if (g, f ) 7→ g ◦ f in C then ((g, g), (f, f )) 7→ (g, g) · (f, f ) = (g ◦ f, g ◦ f ) • unit laws: for (f, f ) we have 1(Y,Y ) · (f, f ) = (f, f ) = (f, f ) · 1(X,X) • associativity: (h, h)·((g, g)·(f, f )) = ((h, h)·(g, g))·(f, f ) = (h, h)·(g, g)·(f, f ) This is isomorphic to the category C itself and also serves to characterize the functor idC . This implies that if F : C → D is taken to be a species of structures c 2018, Indian Institute of Technology Delhi
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then it is equivalently characterized by the corresponding base structured category (F, C, D) with multiple objects. Now we illustrate the notion of multi-object algebraic structures from a slightly different perspective of internal categories. First we recall the notion of algebraic structures within an ambient category or objects internal to some ambient category from [Mac98] and formulate a comparative difference taking the example of a group and a groupoid internal to some fixed ambient category. A Groupoid in a category is a tuple consisting of, 1. G(0) Object of Objects (also called Base Object B), 2. G(1) Object of arrows, 3. d0 Domain map, 4. d1 Codomain map, 5. e Identity (section), 6. m multiplication, 7. i Inverse map.
This 7-tuple is simply reduced to 4-tuple in the special case of a group as denoted in Equation 5.11.
Groupoid: G = (G(0) , G(1) , d0 , d1 , e, m, i); Group:
G = (G(1) , e, m, i)
(5.11)
The base object B of a groupoid, is a general object of the ambient category which introduces the concept of multiple objects or object of objects or non-pointedness and arbitrary connectedness (since these multiple objects will have arbitrary interconnecting arrows). Precisely when this base object becomes a terminal object of the
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ambient category, which means there are no subobjects then we recover the special case of pointed and connected groupoid which is a group.
d0
$
G(1)
G(1)
;B
d0 =d1 !
/
G(0) = T = ? = B
(5.12)
d1
The identity map produces a section in the case of a groupoid which naturally reduces to an element in case of a group as below: /
e
B
G(1)
T
/ G(1)
e
(5.13)
In case of groupoid, the object of composable pair of arrows is precisely the fibered product on the base which reduces to a general product in the special case of the group implying all arrows can be composed. p1
G(2) = G(1) ×B G(1) p0
G(1)
/
/
d1
p1
G(1)
/
G(2) = G(1) × G(1) p0
d0
G(1)
B
G(1) /
!
(5.14)
!
T
The multiplication or the composition map is given below consists of 2 squares which are reduced to triviality in case of a group. G(1) o d0
G(2)
p0
d0
Bo
p1
/
G(1)
m
G(1)
d1
/
d1
B
G(1) o !
To
p1
G(2)
p0
!
/
G(1)
m
G(1)
/
!
(5.15)
!
T
The unit laws for left and right identity in case of a group reduce to usual laws of a group. e×id
B ×B G(1)
d0
/ G(2) o %
m
(1)
G
G(1) ×B B
y
id×e d1
e×id
B ×B G(1)
d0
/ %
G(2) o
m
(1)
G
y
G(1) ×B B
id×e d1
(5.16) c 2018, Indian Institute of Technology Delhi
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The composable triple of arrows for a groupoid is again 2 fibered products in serial which reduce to usual products for a group. G(3) = G(1) ×B G(1) ×B G(1)
/ G(1)
G(2)
G(3) = G(1) × G(1) × G(1)
/ G(1)
d0
/B
d1 ◦p1
G(2)
!
/
!
T (5.17)
The usual associativity axiom for these is G(3) m×id
id×m /
G(2)
G(2)
m
m
/ G(1)
G(3) m×id
id×m /
G(2)
G(2)
m
(5.18)
m
/ G(1)
Hence on introspection we deduce a crucial fact that it is the notion of a nonterminal base object which gives rise to a general concept of multi-object species of structures. Some other examples of such species are presented in Table 5.2. Pointed,Connected,Single Object Monoid Group Monoidal Category 2-group Wreath Group (Leyton’s Transfer) Classical Klein Group Geometries Vector Space
Non-Pointed,Arbitrarily Connected, Multi-object Category Groupoid 2-category 2-groupoid Semi-direct Groupoid (Ehrsemann Transport) Groupoid Geometries Vector Bundle
Table 5.2: Single-object against multi-object species of structures Thus equivalent to multi-object base categories, multi-object species of structures could also be characterized as internal categories in some ambient category. For such multi-object species of structures, the inherent natural structure is best modeled using a base which is a category consisting of many objects. The object upon which this base category C acts is X = qX∈Ob(C) F (X) which is the object of objects (or a coproduct of objects) in D and naturally corresponds to a base object of a given multi-object species. Depending on the kind of action, one can expect to capture the whole or partial structure within the base category while the functor (arrow) represents the structure relative to a base. Note that given a multi-object species c 2018, Indian Institute of Technology Delhi
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of structure it is possible to represent this structure relative to a pointed or singleobject base category. As an example we have already seen earlier in Section 5.3 that a wreath group captures the structure of a set consisting of multiple objects or subsets as a single object set. However such a representation compared to multi-object base category such as semidirect groupoid is highly suboptimal and not naturally matched to its true generative structure (in sense of Leyton’s generative theory [Ley01]) and doesnot provide any insight into how compression occurs in the representation. This was studied in the context of PNG standard in 4. Classically the signal processing community has been utilizing single-object species such as vector spaces and group theoretic representations such as wavelets, gabor, shearlets etc formulated through theory of group frames see [?] and references therein until recently in [MG11] where multi-object structure (groupoid C∗ algebra) was explored for sparse time-frequency representations. This has motivated us to study the deeper implications of using multi-object species such as base structured categories with multiple objects in the base category as signal spaces and its impact on fundamental notions of redundancy, linear independence, true dimensionality and general information analysis which we explored in Chapter 4. Thus most general base perspective is offered by a category which covers all the cases both single as well multi-object. This is further strengthened and generalized using a hierarchy of bases leading to multi-resolution kind of analysis on algebraic structures and corresponding n-category species of structures. From the Tables 5.3, 5.4, 5.5 the reader can intuitively infer that relative to the choice of B, the structure of X is resolved. This gives rise to a crucial question of how does one make appropriate choice of B to be able to get the natural structure within X. Base Category C Singleton (Single Object, Trivial arrow) Set I (Multi-Object, Trivial arrows) Monoid M (Single Object, Multi-arrow) Category C (Multi-Object, Multi-arrow)
Induced structure on X = F (C) in D X modeled as object in D X modeled as coproduct object X modeled as monoid object X modeled as category object
Table 5.3: Structure induced on X in a general category D relative to different bases
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Base Category B Singleton (Single Object, Trivial arrow) Set I (Multi-Object, Trivial arrows) Monoid M (Single Object, Multi-arrow) Category C (Multi-Object, Multi-arrow)
173
Induced structure on X = F (B) in Set X modeled as single whole set X a set partitioned into subsets X as endoset on some set X as small category
Table 5.4: Structure induced on X in Set relative to different bases.
Base Category as Structure Terminal (Single Object, Trivial arrow) Discrete (Multi-Object, Trivial arrows) Monoid M (Single Object, Multi-arrow) Category C (Multi-Object, Multi-arrow)
Induced structure on X = F (B) in Cat X as small 1-category X as coproduct category X as endofunctor category,(strict)Monoidal category X as 2-category
Table 5.5: Structure induced on X in Cat relative to different bases.
In summary, the heuristic conclusions which could be drawn based on the discussion in this section are as follows: • Group theory capturing symmetry structure in applications when seen through the lens of category theory is always relative to a trivial base or precisely in a given ambient category, the terminal object with no sub-objects. Groupoid theory again capturing symmetry could be considered as a generalization of group theory which is generalized to an arbitrary base. The base is precisely any general object containing multiple sub-objects and need not be terminal in a given ambient category. • Groups and more generally single-object species of structures are the algebraic structures formed by collections of transformations of global nature. Breaking global transformations into local transformations leads naturally to multi-object species of structures. Intuitively it can be seen that since group theory (or theory of any single-object species of structure) looks at an object of interest from global or external point of view, to cover all local structural possibilities (essentially when whole object is naturally not rigid relative to parts of it), the theory needs to account for this from a global perspective. The single object
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perspective becomes even more restrictive when the base object consists of internal arrows among sub-objects having structures coming from some external category. To cover such a possibility one needs to a priori start with a reference frame which is in some sense fibered on other reference frame. This crucial dependence of frame of reference on natural object structure makes whole base structured categories more natural. Clearly multi-object base structured category perspective is optimal in all such applied situations where a complex object is naturally evolved from smaller prototype using the concept of Leyton transfer [Ley01] or equivalently Ehresmann transport [Bro87] since it accounts for arbitrary local possibilities quite naturally, coming from its non-pointedness and arbitrary connectedness.
c 2018, Indian Institute of Technology Delhi
Chapter 6
Conclusion and extension
In this work after studying Grothendieck fibration and Leyton’s theory of generativity we combined the intuitions in them to introduce the novel concept of “base structured categories” forming a family of categories that characterizes a functor in various ways in a unique way: • (F, C, D) category characterizes a functor as a ‘category structure preserving morphism’ or a graph of a functor. R ¯ op stems from the perspective of a functor as a multi-object • X oF C or ( Cop F) abstract left category action. R ¯ stems from the perspective of a functor as a multi-object abstract right • Cop F category action. • X oF C stems from the perspective of a functor as a multi-object concrete left category action. R ¯ stems from the perspective of a functor as a multi-object concrete right • Cop F category action.
We proved in Chapter 3 that all of these categories are concretely isomorphic to each other yet only abstractly to the base and signify the fact that the arrows explicitly characterize the base structure since the vertical arrows of the total category are just the trivial identities. This gave rise to a distinct concept of trivial categorification and a new perspective for utilizing the potential of functor relating underlying generative cause with observed concrete effect as a kind of system in some fundamental applications especially signal representation and symmetry which hitherto have
176
been treating objects of D purely in a set theoretic way. The set-theory can be used at the object or local level while category theory is lurking at global level. Next in Chapter 4, Table 4.1 summarized the comparison between classic signal representation framework with the proposed functorial framework. By modeling a source with memory as a groupoid in tune with generative intuition we seek to capture isomorphic relationships between waveforms generated by the source directly impacting the amount of perceived information in signal. The memoryless source having no interdependencies of successive messages is modeled as a discrete category. In functorial framework, the relative perspective offered by category theory provides an authentic tool to model interdependence between sub-signals. This leads to arrow-theoretic structural definition of redundancy resulting into understanding of compression in a natural category theoretic way. The novel concept of using set theory alongside category theory was utilized in the proposed functorial framework for signal representation. We applied the new framework in explaining the underlying mechanics of compressive performance of PNG standard or general lossless differential encoding standards than classic representation techniques in certain cases (e.g. iconic images) which contain a lot of isomorphisms in neighboring pixels or in other word maximal transfer of Leyton’s generators. Finally Chapter 5 expanded the base structured category X oF C and X oF C, developing the representation perspective of a functor as a multi-object category action. In summary: • First we looked at how Leyton’s generative theory of shape could be more faithfully modeled using hierarchy of base structured categories using multiobject groupoid bases as compared to groups. • Next using simple example of permutation action on a finite set at two levels (local and global) we showed that hierarchy of base structured categories is precisely the special case of the theory of composite Grothendieck fibrations. • Further we utilized the notion of 2-group action on a category to propose a hierarchy of 2-groups capturing the symmetry of hierarchy of base structured categories naturally connecting with the higher category theory. • Finally we proposed a simple Definition 105 of groupoid structured geometry and briefly commented on multi-object species of structures in anticipation of c 2018, Indian Institute of Technology Delhi
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the impact of such structures in the applied context of multi-object generative signal spaces.
At the end of this thesis we briefly hint at the possible potential of unifying existing signal representation techniques which can be considered as authentic extension of present work. By variation of base category to include graph with edge composition or topology through OpnX or faces of simplical complex instead of measure and varying codomain category to Vect instead of Riesz appropriately it should be possible cast the work of [Rob13], [DNF+ 13] as sort of special cases of functorial framework in near future. The case of treating the Weyl Hiesenberg or affine Group as base category and codomain category as Hilb covers the Gabor or wavelet signal representation in L2 (R) as studied in classic paper [HW89]. Further the isomorphisms in these categories could potentially model other types of structural redundancies in the similar fashion we used isomorphisms of measurable spaces. Finally we summarize the big picture of the work presented in this thesis as below: • Associates the concept of a functor with the generative cause underlying real world observed effect particularly for signals but in general for various natural phenomena. • Collapses the dual argument of set-theory versus category-theory demonstrating that both can benefit applications existing in harmony together by introducing novel concept of base structured categories. • The proposed framework carries the potential for unifying existing models of structured signal representations and explaining the underpinning of redundancy and compression.
c 2018, Indian Institute of Technology Delhi
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Appendix A
Some results on partial categories
A.1
Partial Categories
Definition 107 Restriction Category [CL02]. A restriction structure on a category R consists of an operator (·) on morphisms which maps each f : X → Y to f¯ : X → X (termed restriction idempotent of f ) such that 1. f ◦ f¯ = f for all f : X → Y , 2. f¯ ◦ g¯ = g¯ ◦ f¯ whenever dom(f ) = dom(g), 3. f ◦ g¯ = f¯ ◦ g¯ whenever dom(f ) = dom(g), ¯ ◦ f¯ = h ◦ f ◦ f whenever dom(f ) = dom(g), 4. h
Such a category is termed as restriction category. In Par(LocMeas, M) the restriction idempotent f¯ : X → X for a partial measurable function f : X → Y is given by the partial identity function f¯(x) = x wherever f is defined on the measurable space X and undefined otherwise. Definition 108 Restriction functor[CL02]: A functor F : R → R0 between restriction categories is a restriction (preserving) functor if F (f ) = F (f¯) for every f ∈ R.
A.1 Partial Categories
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The notion of restriction functor as the term suggests, captures the notion of structure (restriction) preserving morphisms inside the category rCat of restriction categories and restriction functors. Observe that if f¯ = 1X then it is precisely the total map. Total[Par(LocMeas, M)] which is the total subcategory of the argument restriction category embeds via a faithful restriction functor inside Par(LocMeas, M); and is same as the category LocMeas. The notion of restriction category is a useful notion since the related concepts of inverse and dagger categories are easily understood through this notion.
Definition 109 [CL02] Partial Isomorphism: A morphism f : X → Y is called partial isomorphism in a restriction category if there is is a unique morphism f ◦ : Y → X (the partial inverse of f ) such that f ◦ f ◦ = f ◦ and f ◦ ◦ f = f .
Definition 110 [CL02] Inverse category: A restriction category with partial isomorphisms as the only morphisms is termed inverse category.
In Par((LocMeas, M), M), every map is a partial isomorphism making it an example of inverse category. Intuitively inverse categories are ‘groupoids with partiality’ when compared with restriction categories which are ‘categories with partiality’.
Definition 111 [Heu13] Dagger category: A category C is a dagger category if it is equipped with a contravariant endofunctor † : C → Cop which is identity on objects and an involution † ◦ † = idC . In other words, †(1X ) = 1X † = 1X for all objects X and † ◦ †(f ) = f †† = f for all morphisms f in C.
c 2018, Indian Institute of Technology Delhi
LIST OF PAPERS BASED ON THESIS 1. S. Samant., “Fibred signal representation", Abstract presented at the International Category Theory Conference CT, 2016. 2. S. Samant.,S. D. Joshi., “Functorial Signal Representation: Foundations and Redundancy", Selected in Proceedings of IEEE Twenty Fourth National Conference on Communications (NCC) 2018. 3. S. Samant.,S. D. Joshi., “Unified Functorial Signal Representation I: From Grothendieck fibration to Base structured categories.", arXiv:1610.05926 4. S. Samant.,S. D. Joshi., “Unified Functorial Signal Representation II: Category action, Base Hierarchy, Geometries as Base structured categories.", arXiv:1611.02437 5. S. Samant.,S. D. Joshi., “Unified Functorial Signal Representation III: Foundations, Redundancy, L0 and L2 functors.", arXiv:1710.10227
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Brief Biodata of Author Salil Samant received the B.E. degree in Electronics Engineering from University of Mumbai., M.Tech. in Communications Engineering from IIT Delhi, India in 2004, and 2007 respectively. He is currently pursuing the Ph.D. Degree in Electrical engineering at IIT Delhi. His research interests include signals and systems using category theory. From 2007 to 2009, he was with Reliance Communications Pvt. Ltd. as a Deputy Manager. From 2010 to 2011 he worked as an implementation and data analysis engineer, with Tekelec Systems Pvt. Ltd (acquired by Oracle Corporation in 2013).
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