LOCALISATION AND QUOTIENT CATEGORIES ALAMEDDIN, ANWAR

UNIVERSITY OF LIVERPOOL, UK

Contents 1. Localisation of Categories 1.1. Category of Fractions 1.2. Calculus of Fractions 2. Quotient Category 3. Relation between Localisation and Quotienting 4. Symmetric Monoidal Categories 4.1. Internalisation 4.2. Enriched Categories References

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Different notions in mathematics are invariant with respect to some classes of morphisms between the studied objects, for instance most algebraic topological invariants are invariant with respect to homotopy equivalences. In such situations, the homotopy category with respect to that class of morphisms becomes the category of main interest, as the natural framework to consider such notions. The homotopy category of a pair (C, S) of a category C and a class S of its morphisms is, roughly, the ‘minimal’ category obtained from C in which morphisms of S are inverted. The homotopy category of the pair (C, S) is referred to by the localisation of C with respect to S. On the other hand, when one is interested in studying mathematical morphisms up-to certain equivalent relation between parallel morphisms, it is natural to identify the related morphisms. That gives rise to the quotient category with respect to the relation in consideration, as a ‘minimal’ category that identify related morphisms. For instance, identifying homotopic continuous maps between topological spaces gives rise to the classical homotopy category of topological spaces, not to be confused with the homotopy category of topological spaces. The former is the quotient category of topological spaces with respect to homotopy relation, whereas the latter is the localisation of the category of topological spaces with respect to homotopy equivalences. Although the two notions of localisation and quotienting are priorly different, it turns out that they are related, and in ‘nice’ situation they give rise to equivalent categories. For instance the classical homotopy category of CW -complexes is equivalent to the localisation of CW -complexes with respect to weak homotopy equivalences. In this file, we recall the notions of localisation, and quotienting, their relations, and related notions. This treatment is mainly based on [GZ67], [GM03], [KS06], [ML98], and [Sul05] . 1. Localisation of Categories Localisation of categories is motivated and was given rise to by the more traditional notions of localisations in algebra and topology, [Whi11], and it is useful to first recall these notions. However, eager readers can skip to localisation of categories. Recall that in different branches of mathematics, and particularly in algebraic geometry, some questions can be considered locally at prime ideals (points of schemes). Then, one can ask the local-to-global question if the local solutions can be glued together into a global solution. Problems that can be attacked this way are said to Date: March 30, 2016. 1

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ALAMEDDIN, ANWAR

UNIVERSITY OF LIVERPOOL, UK

satisfy the local-to-global principle, also known as Hasse principle. The part of the story that we are concerned with here is the local consideration, known as localisation. Localisation of Rings. Let R be a unitary ring, and let S be a multiplicative subset in R. Then, the localisation of R away from S, if exists, is an initial universal morphism of unitary rings ϕS ∶ R → R[S −1 ] that sends elements of S to invertible elements. To be more precise, call an R-algebra R → A S-local if the structure morphism send elements of S to invertible elements in A. Then, ϕS is a universal morphism from idR ∶ R → R to the inclusion of the full subcategory of S-local R-algebras in R-Alg, if exists, see [ML98, §.III.1]. Notice that the definition of initial universal property translates precisely to the requirement of R[S −1 ] being S-local and ϕS being S-equivalence, where a morphism of R- algebras f ∶ A → B is called an S-equivalence if the pre-composition with f induces a bijection f ∗ ∶ R-Alg(B, C) Ð→ R-Alg(A, C) for every S-local R-algebra C. Recall that the category of (small) unitary rings and their morphisms is equivalent to the category of (small) Ab-monoids and their additive functors Ring Ð→ Ab-Mon.

A unitary ring R is sent to the Ab-monoid R with one object ∗ and the ring of endomorphisms EndR (∗) = R, i.e. every element x of R is an endomorphism x ∶ ∗ → ∗ in R. The composition of morphisms in R is given by multiplication of the original ring, whereas the Ab-enriched structure of R is given by the underlying abelian group structure of the ring R. Then, a morphism x ∶ ∗ → ∗ is an isomorphism in R if and only if the element x is invertible in R. The localisation morphism ϕS ∶ R → R[S −1 ], if exists, is sent by the above equivalent of categorises to an additive functor of Ab-monoids ϕ S ∶ R → R[S −1 ], where R[S −1 ] is the image of the ring R[S −1 ] along the above equivalence. In fact, ϕ S is also a universal morphism, analogue to the one above. To see that, notice that ϕ S is an R ↓ Ab-monoid, i.e. object in in the R ↓ Ab-Mon, let S denote the class of morphisms in R that corresponds to the elements in S, and call an R ↓ Ab-monoid S-local if it sends morphism of S to isomorphisms. Since additive functors between Ab-monoid are completely determined by the ring morphisms between the rings of endomorphisms of their unique object, then ϕ S is a universal morphism from idR to the inclusion of the full subcategory of S-local R ↓ Ab-monoid in the category R ↓ Ab-Mon. Localisation of Modules. Similar, but more interesting, argument applies to localisation of left R-modules M away from a multiplicative subset S ⊂ R, for a unitary ring R, if exists. This localisation is defined to be the initial universal morphism ϕS ∶ M → M [S −1 ] from M to the inclusion of the full subcategory of S-local left R-modules in R-Mod, if exists, where a left R-module M ′ is called S-local if the R-linear maps induced by the action of elements of R x×−∶

M′ m

→ ↦

M′ xm

are isomorphisms for every x ∈ S. In other words, it is an S-equivalence from M to an S-local left R-module M [S −1 ], where an R-linear map f ∶ M → N is called S-equivalence if the pre-composition with f induces a bijection f ∗ ∶ R-Mod(N, L) Ð→ R-Mod(M, L) for every S-local left R-module L. Recall that when R is commutative then the localisation of rings R → R[S −1 ] exists, and it is isomorphic (as R-modules) to the localisation of R away from S as a left R-module. In general, there is a isomorphism of R-modules M [S −1 ] ≅ M ⊗R R[S −1 ] for every R-module M , and under the above assumption that R is

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commutative, there exists a localisation of every left R-module away from M , given by the canonical R-linear map M → M [S −1 ] ≅ M ⊗R R[S −1 ]. The tensor functor − ⊗R R[S −1 ] ∶ R-Mod Ð→ R-Mod sends R-linear maps x ⋅ − ∶ M → M to isomorphism for every x ∈ S and M ∈ R-Mod, and hence its image is contained in the full subcategory of S-local R-modules. Since the full subcategory of S-local R-modules in R-Mod is isomorphic to the category of R[S −1 ]-modules, then the above tensor functor factorises into the pre-composition of the inclusion functor US ∶ R[S −1 ]-Mod Ð→ R-Mod. with a functor ϕ S ∶ R-Mod Ð→ R[S −1 ]-Mod. Theorem [ML98, §.IV.1.Th.2] implies that the functor ϕ S is a left adjoint of US , and hence it is a reflector of R-Mod in R[S −1 ]-Mod. Notice that the components of the unit of this adjunction are isomorphic to ϕM for every M ∈ R-Mod. Since R[S −1 ] ⊗R R[S −1 ] ≅ R[S −1 ] then the adjunction monad US ϕ S = − ⊗R R[S −1 ] is 2-idempotent. Also, since R[S −1 ] is a flat module over R, i.e. − ⊗R R[S −1 ] is exact, then ϕ S is also an exact functor. Call a category under R-Mod S-local if the R-linear maps (x ⋅ −)M ∶ M → M are sent to isomorphisms by the structure functors for every M ∈ R-Mod and x ∈ S. Since US is the inclusion functor, and the tensor functor − ⊗R R[S −1 ] sends (x ⋅ −)M to isomorphisms for every M ∈ R-Mod and x ∈ S, then so does the functor ϕ S , and hence ϕ S ∶ R-Mod → R[S −1 ]-Mod is an S-local category under R-Mod. In fact, it is 2-universal S-local category under R-Mod, and is called the localisation functor of R-modules away from S, and it sends an R-module M to its localisation M [S −1 ] ≅ M ⊗R R[S −1 ] away from S. Observe the difference between the two cases of localisation of rings and localisation of R-modules. On contrary to the case of rigs, the multiplicative subset is fixed for the base ring and hence for all its modules, then the above localisation functor localises all R-modules at ones, when it exists. Also, notice that for any monoidal functor F ∶ R-Mod → C into a monoidal category C, F sends the R-linear maps (x ⋅ −)M to isomorphism for every x ∈ S and M ∈ R-Mod if and only if it sends R-linear maps (x ⋅ −)R to isomorphism for every and for every x ∈ S. To see that, recall that the monoid of endomorphisms of the unitary object of a monoidal category is commutative and acts on the monoids of endomorphisms of other objects in the category. R is the unit object of R-Mod, and for every f ∈ EndR-Mod (R) and M ∈ R-Mod, one has an endomorphism M ⊗f ∶

M

ρM

/ M ⊗R

idM ⊗f

/ M ⊗R

ρ−1 M

/ M.

In fact (x ⋅ −)M ∶ M → M coincides with M ⊗ (x ⋅ −)R . Since left unit morphisms ρM is an isomorphism for every M ∈ R-Mod, then the monoidal functor F sends (x ⋅ −)M to isomorphisms for every x ∈ S and M ∈ R-Mod if and only if it sends (x ⋅ −)R to isomorphisms for every x ∈ S. Localisation of Topological Spaces. A similar theory of localisation for ‘nice’ (pointed topological homotopy) types was developed by Sullivan in 1970, see [Sul05]. All the homotopy groups, homology groups, generalised cohomology groups and other algebraic invariants of some ‘nice’ types are naturally educed with abelian group structures, i.e. Z-modules. Then, those invariants can be localised away from multiplicative subsets of Z. Localising those algebraic invariants simplifies some problems in homotopy theory, like computing the homotopy groups of n-spheres. The main idea of localising types is to seek a type that is ‘close enough’ for the given one and whose algebraic invariants are the localisations of those of the original type; and then attack the problem in consideration for the localised types. The words nice and close will be made precise in the below argument. For more on the motivation for the localisation of topological types with respect to homological functors, see for instance the thread of Mike Shulman’s question on mathoverflow.net. The existence of a canonical, unique, abelian group structure on the set of path-connected components of a type depends on the cardinality of the set. The canonical choice of that cardinality that guarantees the existences of a unique such structure is one! Hence, one restricts attention to path-connected types whose fundamental groups are abelian, in order to have all homotopy groups being Z-modules. We will restrict our attention further when recalling Sullivan’s results. However, there exists other constructions of localisation of more general types, like nilpotent spaces. We call types that satisfy the above assumptions ‘abelian’ spaces,

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ALAMEDDIN, ANWAR

UNIVERSITY OF LIVERPOOL, UK

not to be confused with abelian spaces in [Pos77, p.131]. ‘Abelian’ spaces are general enough so the above localisation question makes sense. Denote the full subcategory of ‘abelian’ spaces in hTop by AS. Let l be a set of primes, not necessary non-empty, denote the multiplicative subset of Z generated by prime numbers that are not in l by S, and let Zl ∶= Z[S −1 ]. Then, an ‘abelian’ space is called l-local if all its homotopy groups are Zl -modules, and a morphism f ∶ X → Y in AS is called l-equivalence if the pre-composition with f induces a bijection f ∗ ∶ AS(Y, Z) Ð→ AS(X, Z) for every l-local ‘abelian’ space Z. Then, the l-localisation of X ∈ AS, or the localisation of X at l, is defined to be an l-equivalence ϕ l ∶ X → Xl for l-local ‘abelian’ space Xl , if exists. That is evidently equivalent to ϕ l being a universal morphism from X to the inclusion of the full subcategory of l-local ‘abelian’ spaces in AS. When X is an ‘abelian’ space of a homotopy type of a CW -complex whose fundamental group has a trivial canonical action on the homotopy and homology groups, then the l-localisation ϕ X ∶ X → Xl exists, and Xl has the desired l-localised homotopy and reduced homology groups, i.e. π∗ (Xl ) ≅ π∗ (X) ⊗ Zl and ̃∗ (Xl ) ≅ H ̃∗ (X) ⊗ Zl H ̃∗ (−) is the functor of reduced integral homology groups. In fact, under the above assumptions, having where H either of the above two isomorphisms is equivalent to the morphism X → Xl being an l-localisation of X, see [Sul05, Th.2.1]. ‘Abelian’ spaces that satisfy the above additional assumptions are called ‘simple’ spaces. Denote the full subcategory of ‘simple’ spaces in hTop by SS. For a ‘simple’ space X, the construction of the l-localised space Xl shows that it is a ‘simple’ space. Let Ul be the inclusion of the full subcategory of l-local ‘simple’ spaces SSl in SS. Then, by theorem [ML98, §.IV.1.Th.2], there exists an adjunction Ll ∶ SS ⇄ SSl ∶ Ul with Ll (X) = Xl , ∀X ∈ SS, i.e. the l-localisation is a reflector of SS in SSl ; and the components of the unit of the adjunction η ∶ idSS → Ul Ll are isomorphic to ϕ X for X ∈ SS. Since Zl ⊗ Zl ≅ Zl then the adjunction monad Ul Ll is 2-idempotent. Notice that a morphism f ∶ X → Y in SS induces an isomorphism between l-localised reduced homology groups if and only if fl ∶= Ll (f ) ∶ Xl → Yl induces an isomorphism between reduced homology groups, or equivalently be a homotopy equivalence, i.e. an isomorphism in SS, and hence in SSl . That is due to ϕ X inducing isomorphisms on l-localised reduced homology groups for every X ∈ SS, chasing the unit diagram for a morphism f , and applying theorem [Sul05, Th.2.1]. Recall that for every integer d there exists an endomorphism fn,d ∶ S n → S n in SS that is of degree d for every integer n ≥ 0, see [Hat02, Ex.2.31]. For every prime p ∉ l and integer n ≥ 0, the degree map fn,p induces an isomorphism of localised reduced cohomologies ̃∗ (S n ) ⊗ Zl Ð→ H ̃∗ (S n ) ⊗ Zl . fn,d ∗ ∶ H Then, by [Sul05, Th.2.1], Ll (fn,d ) are isomorphisms in SSl . In fact, Ll is a 2-universal functor sending the degree maps f2,d to isomorphisms for every p ∉ l, see [Far96, §.1.E.2]. Call a category under SS l-S n -local if the structure functor sends the degree maps fn,d to isomorphisms for every p ∉ l, then in particular Ll ∶ SS → SSl is l-S 2 -local. Under the above assumption on SS, a category under SS is l-S 2 -local if and only if it is l-S 1 -local. The above notations and results are based on the cellular construction of localisation introduced in Sullivan’s lecture notes. Since our aim from recalling localisation of topological types here is to motivate the concept of localising categories, we wont continue further with this review. However, it is worth mentioning that different notions of localisation of more general topological types have been developed, and those notions are not always equivalent outside nilpotent spaces, see for instance [MP12, Part.2].

LOCALISATION AND QUOTIENT CATEGORIES

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Localisation of Groups. l-localising an abelian group G, considered as Z-module, away from a multiplicative subset of Z inverts degree maps for prime degrees that are not in l. The degree maps for abelian additive groups as just a special case of the power maps for multiplicative groups. Then, similarly, for a set of prime numbers l, one calls a multiplicative group (not necessary abelian) l-local when the p-power maps are isomorphisms for p ∉ l, and call a homomorphism of groups f ∶ G → H l-equivalence if the pre-composition with f induces a bijection f ∗ ∶ Grp(H, Z) Ð→ Grp(G, Z) for every l-local group. Then, the l-localisation of group G away from l is defined to be a universal morphisms Ll ∶ G → Gl from G to the inclusion of the full subcategory of l-local groups in Grp, if exist. Then, in particular, Ll ∶ G → Gl is an l-equivalence of groups. Also, for a set of prime numbers l, a categories under Grp is called l-local if its structure functor sends p-power maps −p ∶ G → G to isomorphisms for every group G and p ∉ l. Then, the l-localisation functor for groups is defined to be the 2-universal l-local category under Grp, if exists. Localisation of Categories. The localisation of categories generalises the above notions of localisations. The main idea underlying the above arguments is being able to universally invert a class of morphisms in a category associated to the objects in consideration. The eager reader can find the definition in 1.1. Localisation of categories can be given in the language of categories (1-categories). However, in order to see how the definition fits into the bigger picture, and to see how it honestly generalise the above mentioned notions of localisations, one needs to view it as 2-universal 1-morphisms. We start with an argument that ‘naturally’ motivate the definition in terms of 2-universal 1-morphisms, Def.1.1. Then, we expand the definition and recover the more common definition in terms of 1-categories. Let C be a category, S a class of morphisms in C. A functor F ∶ C → D is called S-local if it sends morphisms in S to isomorphisms in D. Then, one priorly wants to define the localisation of C with respect to S to be the initial universal S-local functor LS ∶ C Ð→ C[S −1 ], if exists. Then the ‘category’ C[S −1 ] is then isomorphic to the category of fractions of C with respect to S, recalled below in 1.1. The construction of the category of fractions has several disadvantages. They are mainly due to its size and to the nature of its morphisms, that it might be big enough not to be a category in given foundations, and its morphisms are formal and hard to work with compared to those of C. On the other hand, one rarely obtains isomorphisms of categories in every day mathematics, and it is rather equivalence of categories what one usually in interested in. Since universal morphisms are defined up-to isomorphisms, and strict 2-universal 1-morphisms are defined up-to 1−equivalences. Then, in order to relax the above suggested notion of localisation, and obtain a notion up-to equivalence of categories, on can define the localisation functor LS ∶ C Ð→ C[S −1 ] using to be the initial strict 2-universal S-local functor, if exists. To be precise, strict 2-universal morphisms are defined from objects in a strict 2-category to a strict 2-functor, and hence one needs to realise the property of functors being S-local in terms of the existence of a 1-morphisms to an suitable strict 2-functor. Let CAT2 be the very large strict 2-category of categories, functors, and natural transformations; let iS ∶ S ↪ C be the subcategory of C generated by S, and observe that a functor F ∶ C → D sends morphisms in S - and hence morphisms in S - to isomorphisms if and only if there exists a strict 2-commutative squares S SqF ∶



iS e

−op

 y


∼

op

S

j

/C F

 /D

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ALAMEDDIN, ANWAR

UNIVERSITY OF LIVERPOOL, UK

in CAT2 , i.e. if there exists a natural isomorphism e ∶ F iS ⇒ j−op . That, on the one hand, the existence of such natural isomorphism induces a commutative square eX

F (X)

/ j(X op ) O j(sop )

F (s)

 F (Y )

eY

/ j(Y op )

in D whose horizontal morphisms are isomorphisms, for every morphism s ∶ X → Y in S, in particular for op those in S. That in turn implies that F (s) is an isomorphism with an inverse e−1 X j(s )eY , for every morphism s ∶ X → Y in S. On the other hand, assume that F sends morphisms in S to isomorphisms and define the −1 functor j ∶ Sop → D by j(X op ) = F (X) for X op ∈ Sop and j(sop ) = F (s) for sop in Sop . Then, there exists op a natural isomorphism e ∶ F iS ⇒ j− given by eX = idX for every X ∈ S, and hence the strict 2-commutative squares SqF exists. It is evident that j in SqF is unique up-to natural isomorphism. Let ●←●→● be the span small category, and consider the very large strict 2-category CAT●←●→● of strict 2-functors 2 from ●←●→● to CAT2 , their pseudo-natural transformations, and the modifications of the latter; i.e. objects of ●←●→● to CAT2 are spans of functors of large categories Uo

G

T

H

/V,

its 1-morphisms between objects U ← T → V and U′ ← T′ → V′ are triples of functors (U ∶ U → U′ , T ∶ T → ∼ ∼ T′ , V ∶ V → V′ ) and pairs of natural isomorphisms (φ ∶ U G ⇒ G′ T, ψ ∶ V H ⇒ H ′ T ) Uo U

 U′ o

G

∼

φ

G′

T T

$  z T′

H ψ

∼ H′

/V V

 / V′ ,

its 2-morphisms between 1-morphisms ((U, T, V ), (φ, ψ)) and ((U ′ , T ′ , V ′ ), (φ′ , ψ ′ )) are triples of natural transformations (u ∶ U ⇒ U ′ , t ∶ T ⇒ T ′ , v ∶ V ⇒ V ′ ) that respects evident composition of natural transformations, i.e. H ′ t = ψ ′ ⋅ vH ⋅ ψ −1 , and G′ t = φ′ ⋅ uG ⋅ φ−1 Uo

T

G

/V

H

′

∼φ

U′

Uo

U

AI

V′

BJ

T

G

u

t

 U′ o

∼φ

 U′ o G′ There is a diagonal strict 2-functor

AI

ψ′

T′

∼

ψ

∼

$  z T′

G′

/V

H

V

v

 / V′

H′

T

$  { T′

H′

 / V′ .

∆ ∶ CAT2 Ð→ CAT2●←●→● sending a category D to the span Did←D Did→D D, a functor F ∶ D → E to the triple of functors (F, F, F ) and pair of identity natural isomorphisms (idF , idF ), and a natural transformation e ∶ F ⇒ G ∶ D → E to the triple of natural transformations (e, e, e).

LOCALISATION AND QUOTIENT CATEGORIES

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Notice that a functor F ∶ C → D fits into a strict 2-commutative square SqF if and only if it fits into a op 1-morphism ((G, H, F ), (φ, ψ)) from the span Sop −← Si→S C to ∆(D), and hence F inverts morphisms of S if and only if fits into such 1-morphism. Notice that the choice of G and H in the above 1-morphism to ∆(D), if exists, is unique up-to 2-isomorphisms. However, the choice of φ, ψ is not necessary unique. To ease notion, when ψ = φ = idF iS , and hence H = F iS and G = F iS −op , we refer to the above 1-morphism by F . Definition 1.1 (Localisation of Categories). Let C be a category, S a class of morphisms in C. Then, the localisation of C with respect to S, if exists, is defined to be a strict 2-universal 1-morphism LS ∶ C → C[S −1 ] from the span

−op iS Sop ← S → C

to the strict 2-functor ∆ ∶ CAT2 Ð→ CAT2●←●→●

where iS ∶ S ↪ C is the subcategory of C generated by S. Since MapCAT2 (U, V) ∶= Funct(U, V), then a functor LS ∶ C → C[S −1 ] is the localisation of C with respect to S if and only if the pre-composition with LS induces an equivalence of categories op i S S→C

(Sop −← L∗S ∶ Funct(C[S −1 ], D) ≅ MapCAT●←●→● 2 given by

L∗S (G)

, ∆(D)) ,

= G LS , being for every category D.

The above definition of the localisation LS means in particular that it fits into a strict 2-push-out square of op the span Sop −← Si→S C in the very large strict 2-category CAT2 . Strict 2-colimits are unique up-to 1-equivalence, if exists, and hence the localisation is unique up-to equivalence of categories, if exists. Recall that L∗S is an equivalence of categories if and only if it is essentially surjective and fully faithful. The functor L∗S is essentially surjective if and only if every functor F ∶ C → D that sends morphisms of S to isomorphisms 2-factorise through LS , i.e. factorises up-to natural isomorphism. On the other hand, L∗S is fully faithful if and only if the pre-composition with natural isomorphism idLS induces a bijection of sets op i S S→C

(Sop −← L∗S G,H ∶ Funct(C[S −1 , D)(G, H) ≅ MapCAT●←●→● 2

, ∆(D)) (G LS , H LS ) ≅ Funct(C, D)(G LS , H LS )

for every category D and for every pair of functors G, H ∶ C[S −1 ] → D. That in turn is equivalent to the pre-composition with LS , L∗S ∶ Funct(C[S −1 ], D) → Funct(C, D), being fully faithful for every category D. Therefore, the above definition is equivalent to definition one induces from proposition [GZ67, §.1.3], and to the definition [KS06, Def.7.1.1]. That is the localisation of C with respect to S, if exists, is a functor LS ∶ C → C[S −1 ] such that ● For every functor F ∶ C → D that sends morphisms of S to isomorphisms there exists a functor GF ∶ C[S −1 ] → D and a natural isomorphism φ ∶ GF LS ≅ F . ● The functor given by pre-composition with LS , L∗S ∶ Funct(C[S −1 ], D) → Funct(C, D), is fully faithful for every category D. If the localisation LS ∶ C → C[S −1 ] exists, the by definitions it sends morphisms of S to isomorphisms. However, it may as well send morphisms that are not in S to isomorphisms. We define the saturation of S to be the multiplicative kernel of LS , if exists, i.e. the subcategory of C whose morphisms are precisely morphisms of C sent to isomorphisms by LS . In particular the saturation of S contains all isomorphisms of C. Recall that the class of isomorphisms satisfy the two-out-of-six-property in any category, i.e. given three f

g

h

composable morphisms X → Y → Z → W in the category, if gf, hg are in the class then so is f, g, h, and hgf . It is evident that the saturation of S satisfies the two-out-of-six-property in C. In fact, it is easy to see that the saturation of S is the smallest subcategory of C that contains S and satisfies the two-out-of-six-property in C. Lemma 1.2 (Reflective Localisations). Let C be a category, S be a class of morphisms in C, and LS ∶ C Ð→ C[S −1 ] be a localisation of C with respect to S. If LS admits a right adjoint US , then the following conditions are equivalent

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ALAMEDDIN, ANWAR

UNIVERSITY OF LIVERPOOL, UK

● US is fully faithful. ● The adjunction counit  is a natural isomorphism. Proof. See [GZ67, §.I.Prop.1.3].



Localisations that satisfy either of the equivalent conditions of the above lemma are said to be reflective, as they are reflectors from C[S −1 ] to a subcategory of C. Then, in particular one has C[S −1 ] fully faithfully embedded in C. Not all localisations are reflective in general. However, several interesting localisations are, as seen below. Example 1.3. The localisation of left R-modules away from a multiplicative subset S of R and the localisation of ‘simple’ spaces at a set of prime numbers are reflective, as seen above. Also, the localisations of the category of CW -complexes with respect to weak equivalences, resulting in the homotopy category of CW -complexes, and the localisation of the complexes of abelian category with respect to quasi-isomorphisms are also reflective. Example 1.4. The localisation of homotopy categories derived from Bousfield localisation of model categories is reflective, see [Ala16, Model Categories, Def. 2.8.40]. 1.1. Category of Fractions. Let C be a category, S be a class of morphisms in C. Then, one can always consider the (priorly big) category of fractions of C with respect S. When it exist, i.e. when its hom-objects being sets as opposed to proper classes, it gives a localisation of C with respect S. Caution 1.5. Recall the construction of the category of fractions of C with respect S. It given by the quotient category of the free category F G(C, S −1 ) with respect to the minimal relation R that satisfies: ● ∀X ∈ C, ()X RX,X (idX ). ● (f, g) RX,Z (g ○ f ) for every pair of composable morphisms, f ∶ X → Y and g ∶ Y → Z. ● ∀s ∶ X → Y in S, ()X RX,X (s, s−1 ), and ()Y RY,Y (s−1 , s). It has the same class of object as of C, whereas its morphisms are equivalence classes of zigzags1 of morphisms in C with the components directed backwards being from S, modulo the evident equivalence relation induced from R, see [GZ67, §.I.1.1] for the concrete construction. This construction results in several difficulties: ● The category of fractions of C with respect S might not exist, in the given foundations e.g. the universe C lives in. That, hom-objects do not have to be small sets, and in general they may be proper classes. ● The nature of the morphisms of the category of fractions becomes formal and obscure compared to those of C, even if C is a ‘nice’ well-understood category. The above difficulties can be remedied when the class we are localising with respect to is a set and when the given category admit suable structure, like left or right calculus of fractions, or a model category structure. We review below calculus of fractions, and we devote another file for the study of model structures and their localisation, see [Ala16, Model Categories, §.2]. 1.2. Calculus of Fractions. Recall that, in reflective localisation, the localised category is fully faithfully embedded in the original category. That guarantees its existence in the given foundations (that admit the axiom of choice) and makes it more accessible. Analysing the reflective localisation, isolating the axioms resulting in a such good behaved localised category, as done in [GZ67, §.I.2], one distinguished the below desired properties of the class one is localisation with respect to. Definition 1.6 (Calculus of Fractions). Let C be a category, S a set of morphisms of C. Then, C is said to admit calculus of left fractions with respect to S, and S is said to be a left multiplicative system in C if the following axioms hold: (1) S is a subcategory of C that contains all its objects, i.e. the identity of every object of C is in S and the composition of morphisms of S is in S. (2) (Weak Push-outs) For every solid diagram X

f

s

 X′

/Y s′

f′

 / Y′

in C with s ∈ S, there exists doted morphisms in C completing it to a commutative square with s′ ∈ S. 1Strings from the joint class of morphisms of C and formal inverses of morphisms of S.

LOCALISATION AND QUOTIENT CATEGORIES

9

(3) (Annihilation) For every commutative solid diagram X′

s

/X

f g

// Y

s′

/ Y′

in C with s ∈ S there exists a doted morphism s′ ∈ S making the whole diagram commutative. C is said to admit calculus of right fractions with respect to S, and S is said to be a right multiplicative system in C if the dual axioms hold. Moreover, C is said to admit calculus of fractions with respect to S, and S is said to be a multiplicative system in C if C admits both calculus of left and right fractions with respect to S. When C admits a calculus of left fractions, a construct of the category of left fractions S −1 C can be given by the data: ● The class of objects Ob(S −1 C) = Ob(C). ● For every pair of objects X, Y ∈ Ob(S −1 C), S −1 C(X, Y ) is the set of equivalence classes of S-left roofs (u, α) ′ 7Y g α u Y X for some Y ′ ∈ C, u ∈ C(X, Y ′ ), and α ∈ S, with the equivalence relation being generated by having two S-left roofs (u1 , α1 ) ∶ X → Y1′ ← Y and (u2 , α2 ) ∶ X → Y2′ ← Y equivalent if and only if they admit a corefinement, i.e. if there exists an S-left roof (u, α) ∶ X → Y ′ ← Y and morphisms s1 ∶ Y1′ → Y ′ and s2 ∶ Y2′ → Y ′ in S making the below diagram commutative 4 Y1′ j

u1

α1

s1

X

u

 / Y′ o O

α

Y.

s2 u2

α2

* Y′ t 2

This relation is indeed an equivalence relation due to S being a left multiplicative system in C, especially due to the weak push-outs and annihilation properties, by lemma [GM03, §.III.2.Lem.8]. In fact, one can view S −1 C(X, Y ) as a filtered colimit of S-left roofs, see [Sta16, Tag 04VB]. ● For every triple of objects X, Y, Z ∈ Ob(S −1 C) and S-left roofs (v, β) ∶ Y → Z, and (u, α) ∶ X → Y , applying the axiom of choice, one fix a commutative diagram induced by the weak push-out axiom with α′ ∈ S ′′ 6 Z h α′ v′ u

′ 7Y h

α

v

′ 6Z g

β

X Y Z. Let S-roof (X, Y ) be the class of roofs from X to Y . Then, there exist a class function ○ ∶ S-roof (Y, Z) × S-roof (X, Y ) Ð→ S-roof (X, Z) given by (v, β) ○ (u, α) ∶= (v ′ u, α′ β) for morphisms fitting in the above fixed diagrams. This function extends readily to a well-defined function on equivalence classes, i.e. to a composition function ○ ∶ S −1 C(Y, Z) × S −1 C(X, Y ) Ð→ S −1 C(X, Z). ● For every object X ∈ Ob(S −1 C), let idX ∶ ∗ → EndS −1 C (X) be given by (idX , idX ). One sees readily that S −1 C is a category, see the proof of [GM03, §.III.2.Lem.8]. The construction of S −1 C mimics the construction of the ring of left fraction of a ring away from a multiplicative subset of the given ring. Obviously, the two coincides when C is additive and has one object. One think of the class of S-left roof (u, α) as [α−1 u]. Dual constructions defines the category of right fractions CS −1 . There is an evident functor [−]S ∶ C → S −1 C fixing objects and sending morphisms f ∶ X → Y to classes of S-left roofs [f, idY ].

10

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UNIVERSITY OF LIVERPOOL, UK

Theorem 1.7. Let C be a category, S be a set of morphisms of C, and assume that C admits calculus of left fractions with respect to S. Then, [−]S ∶ C → S −1 C is a localisation of C with respect to S. Proof. See [GZ67, §.I.2.4] and [GM03, §.III.2.Lem.8].



Recall the main problem with the category of fractions is that its morphisms are classes of zigzags −1 −1 −1 s−1 0 f0 s1 f1 s2 f2 ⋯sn fn

of arbitrary length, with s−1 i being a formal inverse of si ∈ S for 0 ≤ i ≤ n. In general one do not have cancellation or commutativity relations that allow better understanding of these formal zigzags in terms of morphisms of the original category. However, the main advantage one obtains when C admits a calculus of left or right fractions with respect to S is that the formal inverses of morphisms of S, appearing in the above morphism, can be always gathered into a single formal inverse on the left or the right, respectively, turning all such zigzags, priorly of arbitrary length, into the relatively better-understood roofs, see [ibid., §.III.2.Rem.7]. 2. Quotient Category One is sometimes interested in steadying mathematical objects up-to some equivalent relation that respect their structure, in such case it is useful to consider the quotient of such objects with respect to the given equivalence relation, when such quotient is defined, as in the case of subsets, normal groups, ideals of rings, submodules, etc. The same principle applies to category theory, for instance, the classical homotopy category of topological spaces hTop = π Top is defined to be the quotient category of topological spaces Top with respect to homotopical equivalence. Hereby, we recall hereby the notions of congruence relation and quotient category. Definition 2.1 (Congruence Relation). Let C be a category. A relation on C is a (class) function R that assigns to each pair of objects X, Y ∈ C a binary relation RX,Y on C(X, Y ) that respects the composition in C, i.e. if f, f ′ ∈ C(X, Y ) and g, g ′ ∈ C(Y, Z) such that f RX,Y f ′ and gRY,Z g ′ , then gf RX,Z g ′ f ′ . A relation R on C is called a congruence relation if all binary relations RX,Y are equivalence relations for every X, Y ∈ C. Let C be a category, R a relation on C, then the argument in [ML98, p.52] shows that there is a minimal ¯ on C with R ⊆ R. ¯ In particular, when R is a congruence relation, then R ¯ = R. congruence relation R Having the notion of homotopy and homotopy equivalence for topological spaces in mind, on generalises these notion as follows: Definition 2.2. Let C be a category, and R a relation on C. Let f, g ∶ X → Y be morphisms in C, f is said to be R-related to g if f RX,Y g, for short one writes f ∼R g. Also, for objects X, Y ∈ C, X is said to be R-equivalence to Y if there exist morphisms f ∶ X → Y and g ∶ Y → X such that gf ∼R idX and idY ∼R f g. Notice that when R is a congruence relation then R-equivalence is an equivalence relation. Definition 2.3. Let C be a category, R be a relation on C. Call a category D under C R-congruent if the structure functor F ∶ C → D identifies R-related parallel morphisms of C in D, i.e. for every X, Y ∈ C and f, f ′ ∈ C(X, Y ) such that f RX,Y f ′ one has F (f ) = F (f ′ ). Then, the quotient category of C with respect to R, if exists, is defined to be to be a universal morphism QR ∶ C Ð→ C/R from idC to the inclusion of the very large full subcategory of S-congruent categories under C in the very large category C ↓ CAT. That is equivalent to QR ∶ C Ð→ C/R being R-congruent category under C, and QR being R-congruent equivalence, where a functor F ∶ D → E between categories under C is called R-congruent equivalence if the functor F ∗ ∶ C ↓ CAT(E, Z) Ð→ C ↓ CAT(D, Z) is an isomorphism between functor categories, for every R-congruent category Z under C. One notices that the above definition is strict, i.e. taken in the very large 1-category CAT. Of course, it makes sense to define its weaker version as it is the case for localisation. However, for an arbitrary category C and a relation R on C, proposition [ibid., Prop.§.II.8.1] implies that the quotient QR ∶ C Ð→ C/R exists, in the strict sense, and hence one does not need to weaken the definition. The argument on page 52 loc.cit. shows ¯ is isomorphic to the category C/R that is given by that C/R ≅ C/R ● A class of objects Ob(C/R) = Ob(C).

LOCALISATION AND QUOTIENT CATEGORIES

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● For each pair of objects X, Y ∈ Ob(C/R), the set of morphisms C/R(X, Y

) ∶= C(X, Y )/R¯ X,Y .

¯ X,Y . is the set-theoretical quotient with respect to the minimal equivalence relations RX,Y ⊆ R ● For each triple of objects X, Y, Z ∈ Ob(C/R) the composition maps of C in the first row induces a unique composition map in C/R in the second row that R respects the composition of C ○C ∶

○C/R ∶

C(Y, Z) × C(X, Y )

/ C(X, Z)



 / C/R(X, Z).

C/R(Y, Z) × C/R(X, Y

)

● For each object X ∈ Ob(C/R), the equivalence class [idX ] of idX in C gives the identity of X in C/R, i.e. the composition of the identity element in C and the quotient morphism gives the identity at X in C/R idX

∗

/ C(X, X)

/ / C/R(X, X).

Example 2.4. Recall that a commutative ring R can be though of as a pre-additive monoid R with one object ∗ and a commutative ring of endomorphisms EndR (∗) = R. Let I ⊂ R be an ideal in R, then the relation on ∼I on R defined x ∼I y if and only if x − y ∈ I defines a congruence relation I on R. The quotient category R/I is isomorphic to the the pre-additive monoid R′ with one object ∗ and a commutative ring of endomorphisms EndR′ (∗) = R/I. On the other hand, let I be a congruence relation I on R that respect its pre-additive structure, then I ∶= {x ∈ R∣x ∼I 0} is an ideal in R, and the quotient category R/I is isomorphic to the the pre-additive monoid R′ with one object ∗ and a commutative ring of endomorphisms EndR′ (∗) = R/I. Notice that a relation R on a category C for which idX R idX for every X ∈ C can be realised as a subcategory (i1 ,i2 )

R ↪ C × C with f ∼R g if and only if (f, G) is in R. Then, ● R is reflexive if and only if there exists a section functor s ∶ C → R for both of i1 and i2 , i.e. (i1 , i2 )s = (idX , idX ). ● R is symmetric if and only if there exists a symmetric braiding functor ψ ∶ R → R that interchanges i1 and i2 , i.e. (i1 , i2 )ψ = (i2 , i1 ). ● R is transitive if and only if there exists a ‘multiplication’ functor µ ∶ R ×C R → R, where the fibre product is given by the Cartesian square p1

R ×C R ⌜ p2

 R

/R i2

i1

 / C,

such that (i1 , i2 )µ = (i1 p1 , i2 p2 ). The first two equivalences are obvious. For transitivity, on the one hand let f ∼R g and g ∼R h in C, then ((f, g), (g, h)) is a morphism in R×C R that i2 ((f, g)) = i1 ((g, h)). Assume the existence of such ‘multiplication’ functor µ, then (f, h) = µ ((f, g), (g, h)) in R, and hence f ∼R h. The other direction is evident. Moreover, the quotient category C/R is given by the coequalizer of the solid diagram R

  (i1 ,i2 ) /

C×C

π1 π1

// C

QR

/ C/R.

in the very large category CAT. Then, the the quotient category C/R is in particular a quotient object in CAT. One can generalise the notion of a relation and define a relation on a category C to consider arbitrary subcategories (i1 , i2 ) ∶ R ⊂ C × C, and define categorical congruence relation on C to be a categorical relation R ⊂ C × C that is reflexive, symmetric, and transitive in the above sense. Then, for a categorical relation R on C, define the categorical quotient of C with respect to R to be the coequalizer of the solid diagram R

  (i1 ,i2 ) /

C×C

π1 π1

// C

QR

/ C/R.

12

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in the very large category CAT. Since the functor (i1 , i2 ) is a mono, then the functors s, ψ, and µ for which (i1 , i2 ) is reflexive, symmetric, and transitive in the above sense are unique, if exists. Hence, a congruency is a property on categorical relation, as opposed to a structure. The above notion of ‘categorical’ relations can be generalised further to any category C with products and (i1 ,i2 )

fibre product. That an internal relation on an object X ∈ C is defined to be a subobject R ↪ X × X. An internal relation R on X is called an internal congruence relation if it is reflexive, symmetric, and transitive, where ● R is reflexive if there exists a section morphism s ∶ X → R in C for both of i1 and i2 , i.e. (i1 , i2 )s = (idX , idX ). ● R is symmetric if there exists a symmetric braiding morphism ψ ∶ R → R in C that interchanges i1 and i2 , i.e. (i1 , i2 )ψ = (i2 , i1 ). ● R is transitive if there exists a ‘multiplication’ morphism µ ∶ R ×X R → R in C, where the fibre product being given by the Cartesian square p1

R ×X R ⌜ p2

/R i2

 R

i1

 / X,

such that (i1 , i2 )µ = (i1 p1 , i2 p2 ). Then, in particular the above ‘categorical’ congruence relation is an internal congruence relation in the very large category CAT. 3. Relation between Localisation and Quotienting Although the notions of localisation and quotienting of categories are priorly different, they are pertained. Their relation fades in some occasions and embellishes in others. The main similarity between the two construction is that both constructions result in inverting a class of morphisms, and both of them are ‘colimits’. Localisations inverts morphisms of a given class, and by doing so it identifies the domain and codomain of morphisms of S, of course up-to isomorphism. Whereas quotienting is a ‘finer’ construction, that also invert the saturated class R-equivalences, however its effect is not restrict to that. Quotient Categories. We start with the more comprehensible case of quotient categories. The main three advantage of the construction of quotient categories are: ● They always exist. ● Their morphisms are represented by morphisms of the original category, and hence the nature of the resulting morphisms is well understood, at least in comparison to morphisms of the original category. ● They admit unique functors induced from functors on the original categories, that respects their relations. Let C be a category R a relation on C, and QR ∶ C → C/R be the quotient functor of C with respect to R. ¯ ¯ As a result of identifying R-related morphisms in C/R, the quotient functor QR takes R-equivalences in C to isomorphisms in C/R, and hence the quotient functor with respect to R 2-factorises in the very large 2-category ¯ CAT through the localisation functor of C with respect to the class of R-equivalences, if the localisation exists. ¯ Denote the class of R-equivalences in C by WR , when the localisation LWR ∶ C → C[WR−1 ] exists, there exist a functor FR ∶ C[WR−1 ] → C/R, unique only up-to natural isomorphism, and hence there exists a triangle of functors (not necessary commutative) /4 C/R,

QR

C LWR

*

C[WR−1 ]

FR

and a unique natural isomorphism FR LWR ≅ QR . ¯ implies that WR is closed (C, WR ) is a category with weak equivalences. In particular, the transitivity of R under composition, its reflexivity implies that WR contains isomorphisms. One direction of the two other cases

LOCALISATION AND QUOTIENT CATEGORIES

13

¯ being symmetric and transitive. of the two-out-of-three property are evident, whereas the rest results from R ¯ In fact, a morphism in C is an R-equivalence if and only if it is sent to isomorphism in C/R, that equalities ¯ between morphisms in C/R accounts to being R-related in C. The natural isomorphism FR LR ≅ QR implies that morphisms in C that are sent to isomorphism in C[WR−1 ] are also sent to isomorphisms in C/R. Hence ¯ a morphism in C is an R-equivalence if and only if it is sent to isomorphism in C[WR−1 ], i.e. the class WR is saturated. In some occasions the functor FR is an equivalence of categories, as it is the case for quotient category of classical homotopy category of CW -complexes and the localisation of CW -complexes with respect to homotopy equivalences. However, in general the functor FR is not even fully faithful, as we will see in the below. Since the two notions of localisation and quotienting of categories generalise their ring-theoretical counterparts, it is useful to consider examples arising from commutative algebra. Let R be a commutative ring, I ⊂ R be an an ideal in R, R and R/I be the pre-additive monoids corresponding to the rings R and R/I, where I is the congruence relation on R given by f ∼I g if and only if f − g ∈ I, as in example 2.4. Then, a morphism f ∶ ∗ → ∗ in R, i.e. f ∈ R, is an I-equivalence if and only if there exists a g ∈ R such that f g − 1 ∈ I (i.e. if [f ] is invertible in R/I). The set SI ⊂ R of I-equivalences forms a multiplicative system in R and one has a factorisation of ring morphisms and pre-additive 1-monoids, respectively 4/ R/I

QI

R LSI

*

R[SI−1 ]

FI

, and

4/ R/I.

QI

R LI

*

R[SI−1 ]

FI

Notice that when I is a maximal ideal in R then SI = R ∖ I, and hence R[SI−1 ] ≅ RI . Then, the the morphism FI is not an isomorphism in general as it is the case for F(2) ∶ Z(2) ↠ Z/2, and hence FI is not an equivalence of categories in general. Avoid the confusion, in general R ∖ I is not necessary a multiplicative system, and even when it is, i.e. when I is a prime ideal, SI and R ∖ I do not have to coincide, and R/I does not have to factorise through RI . For example, let k be a filed, R = k[x, y], and I = (x) ⊂ R be the prime ideal generated by x. y is invertible in RI , but is not invertible in R/I, hence the quotient morphism QI ∶ R → R/I does not factorises through RI . Localisation of Categories. On the contrary to quotient categories, localisation of categories is defined as a strict 2-colimit up-to equivalence of categories and has the following three disadvantages: ● They do not always exist. ● Their morphisms are represented by zigzags of morphisms of the original category, and hence the nature of the resulting morphisms is very formal compared to morphisms of the original category. ● They do not always induce canonical functors from functors on the original categories that respects the localised classes. One approach to remedy some of these disadvantages is given by calculus of fractions, when it is admitted by the localised class, see 1.2, where zigzags can be simplified and be replaced by roofs. Another approach follows from Quillen’s development of homotopical algebra. For a category C and a class of morphisms W ⊂ C, one looks for extra structure M on C, namely model structure [Ala16, Model Categories, Def. 2.1.23], which guarantees the existence of the localisation C[W −1 ] ≅ H(C, M). This extra structure induces a congruence relation H on a full subcategory Cf c ⊂ C (namely, the full subcategory of fibrant-cofibrant objects of (C, M)). Moreover, the existence of such structure implies that the quotient of Cf c /H is equivalent to the desired localisation C[W −1 ]. Hence, for such categories, called model categories, the localisation category C[W −1 ] becomes more accessible through quotient categories Cf c /H. In general, for a category C and a class of morphisms W ⊂ B there may not exist a model structure M on C for which C[W −1 ] ≅ H(C, M), and the localised category C[W −1 ] does not have to be equivalent to a quotient of C or its full subcategories with respect to any congruence relation. Model categories are (co)complete by definition, hence when C is not (co)complete one may use the Yoneda embedding to embed C in the (co)complete category of pre-sheaves on C, or a suitable subcategory of the latter, and then look for a model structure for the ambient category.

14

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UNIVERSITY OF LIVERPOOL, UK

Counter Example 3.1. Using the notation of example 2.4 for R = Z, S = Z ∖ (2) is a multiplicative system in R, but Z(2) is not isomorphic to the quotient of R with respect to any ideal of Z, and hence R[S −1 ] is not equivalent to R/I for any ideal I ⊂ R. However, that also can be remedied easily when on consider the faithful (but not full) embedding R ↪ R′ , where EndR (∗) = R and EndR′ (∗) = R[x], then for the multiplicative system S = {an ∣n ∈ N} ⊂ R, for a non-zero divisor a ∈ R, one has R[S −1 ] ≅ R′ /I where I is the congruence relation defined on R′ by f (x) ∼I g(x) if and only if f (x) − g(x) ∈ (ax − 1). 4. Symmetric Monoidal Categories Symmetric monoidal structures on categories generalises tensor product on vector spaces to arbitrary categories. Roughly speaking, it is a bifunctor on a category satisfying the usual associativity, unitary, and commutativity axioms upto isomorphism, see [ML98, Ch.VII§.1 and §.2] for the definition and basic properties. An object X ∈ C is said to be ⊗-dualisable, if there exists an object X ∗ ∈ C such with morphisms η ∶ 1 → X ⊗X ∗ and  ∶ X ⊗X ∗ → 1 called unit and counit, respectively. X will be simply called dualisable if no confusion arises. Moreover, if η and  are isomorphisms then X is said to be ⊗-invertible, or simply invertible, and then we denote X −1 ∶= X ∗ . Hence, X is invertible if and only if the induced suspension functor ΣX ∶= X ⊗ − ∶ C → C is an endoequivalence of categories. Therefore, X is invertible if and only if ΣX is invertible in the 2-category of categories, equivalence of categories and natural transformation. Euler characteristic. Let X be a dualisable object in C, f ∶ X → X an endomorphism. Then, trace f is defined to be the 1

f ⊗idX ∗

/ X ⊗ X∗

η

/ X ⊗ X∗

≅ ψX,X ∗

/ X∗ ⊗ X



/1

and is denoted by tr(f ). Then, the Euler characteristic of X is defined to be the trace of the identity morphisms idX and is denoted by χ(X), i.e. the composite 1

η

/ X ⊗ X∗

/ X∗ ⊗ X

≅ ψX,X ∗



/1

For motivation and examples see [PS14]. 4.1. Internalisation. Traditional structures (not necessary first-order structures) like group, R-module, ring, topological space structures are set-theoretical, i.e. defined for a set theory. Since the representable functors embed a given (small) category in the category of (small) sets, then one can use these functors to ‘internalise’ the traditional structure into the given category. Let S be the category whose objects are sets with a traditional structure S, and whose morphisms are structure preserving maps, then there is a forgetful functor U ∶ S → Set. Let C be a category, and X ∈ C. Then, the object X is said to be an S-object, or to admit an internalised S-structure if the representable functor hX ∶ Cop → Set factorises through S. Example 4.1 (Group Objects). A group object in a category C is an object X for which the representable functor hX factorises through Grp, / 6 Set.

hX

Cop h′X

(

Grp

U

Since h′X (Y ) admits traditional group structure for every Y ∈ C, h′X is a functor, and that hX (Y ) × hX (Y ) ≅ hX×X (Y ) when C has finite products, then it readily seen that X is a group object in a category C with finite products if and only if there exist morphisms µX ∶ X × X → X

,

ιX ∶ X → X

, and

ηX ∶ ∗ → X

for which µX , ιX , and ηX satisfy the traditional group structure axioms, see [KS06, §.8.1]. Similar argument applies for monoid, commutative group, ring, or R-module objects, etc. For instance, all groups are group objects, or admit an internalised group structure, in the category of groups. A (commutative) monoid object in the category of abelian groups is a (commutative) unitary ring, and a (commutative) monoid object in the category of R-modules is a (commutative) R-algebra.

LOCALISATION AND QUOTIENT CATEGORIES

15

When all objects of a category admit an internalised stricture S, then it is natural to look for such categories where composition maps respect the traditional structure S. For instance, when S is the structure of abelian groups, then the categories whose all objects admit an internalised stricture S and whose composition maps respect the traditional structure S are called pre-additive categories, and they are recalled in [Ala16, ‘Linear Homotopical Algebra’ aka Homological Algebra, Section 1]. 4.1.1. Internalisation of Algebraic Theories. Recall that algebraic objects like (commutative) group, ring, R-module objects can given in categories with finite Cartesian products by the existence of morphisms between suitable Cartesian powers satisfying the traditional structures axioms. Cartesian product is in particular a symmetric monoidal structure, and the conditions the algebraic objects can by given for any symmetric monoidal structure by the existence of morphisms between suitable symmetric monoidal powers satisfying the traditional structures axioms. Notice that algebraic objects given for arbitrary symmetric monoidal structure do not have to coincide with the algebraic objects arisen from the above definition. 4.2. Enriched Categories. The sets of morphisms between objects may admit a natural mathematical structure, usually arising from the structure of the objects of the given category, when they are set-theoretical. For instant, in the category of groups Grp, for every pair of groups G, H ∈ Grp, the set Grp(G, H) has a natural group structure. The same applies for the category of vector spaces Vectk over a field k, and left R-modules R-Mod, for a ring R. Also, in topological spaces, if we restrict ourself to compactly generated Hausdorff spaces, then for any such pair of spaces X, Y , the set CGHaus(X, Y ) admits a canonical topology, making it into topological space, namely the open-compact topology. In all these examples the composition maps ○ ∶ C(Y, Z) × C(X, Y ) → C(X, Z) are compatible with the induced structure, i.e. they are morphisms in the same category. This observation gives rise to enriched categories, where one generalises the notion of category replacing hom-sets with more interesting hom-objects. Definition 4.2 (Enriched Categories). Let (M, ⊗, 1, α, l, r) be a (strict/lax) monoidal category. Then, a small M-enriched category C is given by ● A class Ob(C), called the class of objects in C. ● For each pair (X, Y ) ∈ Ob(C) × Ob(C) of objects of C, an object C(X, Y ) ∈ M, called the hom-object from X to Y . ● For each triple (X, Y, Z) of objects in C, a morphism ○X,Y,Z ∶ C(Y, Z) ⊗ C(X, Y ) → C(X, Z) in M, called the composition morphism. ● For each object X ∈ Ob(C), a morphism idX ∶ 1 → C(X, X), called the identity ‘element’ of X. such that composition in C is (strict/lax) associative, and (strict/lax) unital. Then, C is said to be an M-category. Example 4.3. Any category is a Set-enriched category. A comparison between the definition of categories and enriched categories shows the naturality of the generalisation, and highlights the need to the restriction on M to be (strict/lax) monoidal. Example 4.4. The category of compactly generated Hausdorff spaces is Top-enriched category. Remark 4.5. When M is concrete, i.e. its objects are sets with a traditional structure S, it admits a forgetful functor U ∶ M → Set, and then all objects of C has an internalised structure S. This is the case in the above mentioned examples of enriched categories. In such case, the enriched categories are in particular categories, that there hom-objects are sets equipped with some traditional structure. However, that does not have to hold in general. When the enriched category C happens also to be a category, we distinguish hom-sets from hom-objects, where we denote the hom-set morphisms from X to Y in C by C(X, Y ), and the hom-object from X to Y by M apC (X, Y ), or M ap(X, Y ) if no confusion arises. Definition 4.6 (Enriched Functors). A (strict/lax) enriched functor F ∶ C → D between (strict/lax) M-enriched categories is given by ● A class function F ∶ Ob(C) → Ob(D). ● For each pair (X, Y ) ∈ Ob(C) a morphism FX,Y ∶ Map(X, Y ) → Map(F (X), F (Y )) in M that (strictly/weakly) commutes with the composition and the identity.

16

ALAMEDDIN, ANWAR

UNIVERSITY OF LIVERPOOL, UK

4.2.1. Strict 2-Categories. Recall that the category of small categories Cat is a strict monoidal category. Enriched categories and functors over the strict category Cat are called strict 2-categories and strict 2-functors, respectively. They can be given by classes of objects (0-morphisms), 1-morphisms that satisfy the strict associativity and identity axioms, and 2-morphisms that satisfy the interchange law. The large category Cat of small categories gives rise to the large strict 2-category Cat2 whose objects are small categories, 1-morphisms are functors between small categories, and 2-morphisms are natural transformations between the latter. Also, the very large category CAT of large categories similarly gives rise to the very large strict 2-category CAT2 of large categories, functors between them, and natural transformations between the latter.

The localisation of categorises is given by an (initial) 2-universal morphism in the very large strict 2-category CAT2 . Therefore, we recall (initial) 2-universal morphisms in strict 2-categories.

(Initial) 2-Universal Morphisms in Strict 2-Categories. To motivate the definition, we first recall (initial) universal morphisms in categories, and study its generalisation to 2-categories.

Let F ∶ C → D be a functor between categories, d ∈ D, then an (initial) universal morphism from d to F is defined to be the initial object in the comma category d ↓ F, if exists. If it exists, it is a morphism ηd ∶ d → F (cd ) for cd ∈ C such that for every morphism f ∶ d → F (c) for c ∈ C there exists a unique morphism gf ∶ cd → c in C such that the below diagram strictly commutes d

ηd

/ F (cd )

f

!  F (c)

F (gf )

cd F

←

gf

∃!

 c.

In other words, it is a morphism ηd ∶ d → F (cd ) that induces a bijection of sets ηd∗ ∶ C(cd , c) → D(d, F (c)) for every c ∈ C. The surjectivity of ηd∗ is equivalent to the existence of the factorisation, whereas its injectivity is equivalent to the uniqueness of the factorisation when it exists. The (initial) universal morphism ηd is unique up-to isomorphism, if exists, and the above factorisation of f is unique for the choice of the universal morphism.

Let F2 ∶ C2 → D2 be a strict 2-functor between strict 2-categories, d ∈ D, then an (initial) 2-universal 1-morphism from d to F2 is a 1-morphism ηd ∶ d → F2 (cd ) for cd ∈ C that induces an equivalence of categories ηd∗ ∶ MapC (cd , c) → MapD (d, F2 (c)) for every c ∈ C. Since a functor in part of equivalence if and only if it is essentially surjective and fully faithful, then ηd is a 2-universal 1-morphism from d to F2 if and only if it satisfies the below two conditions: Es. surjective For every 1-morphism f ∶ d → F (c) with c ∈ C there exists (by essential surjectivity) 1-morphism gf ∶ cd → c in C and a 2-isomorphism θf ∶ F2 (gf )ηd ≅ f ′ Fully faithful For every two parallel 1-morphisms f, f ′ ∶ d → F (c) with c ∈ C and a 2-morphism ψ ∶ f ⇒ f ′ there exists (by fullness) a unique (by faithfulness) 2-morphism ξψ ∶ gf ⇒ gf ′ , for the choice of (gf , φf ) and (gf ′ , φf ′ ), such that ψ = φf ′ ⋅ F2 (ξψ )ηd ⋅ φ−1 f . That, in particular, implies that the choice of (gf , φf ) is unique up-to 2-isomorphism (by faithfulness).

LOCALISATION AND QUOTIENT CATEGORIES

d

/ F2 (cd )

ηd

cd

∼φ

f

f

d

17

{

ηd

/ F2 (cd )

cd

F2 (gf )

gf ∃! up-to 2-isomorphisms

F2 (ξψ ) F2

ψ f

←

∼φ

f′

′

 {

F2 (g f ′ )

+  F2 (c)

+  F2 (c)

ξψ ∃!



gf ′



c

c

Notice that the (initial) 2-universal 1morphism ηd is unique up-to 1-equivalences, if exists, and the above factorisation (gf , φf ) of 1-morphism f is unique up-to 2-isomorphisms, for the choice of the universal morphism ηd , whereas the above factorisation ξψ of 2-morphism ψ is unique, for the choice of the factorisation (gf , φf ) and (gf ′ , φf ′ ) of the involved 1-morphism f and f ′ . References ˜ [Ala16] Alameddin, A. 2016. Accessed from: http://pcwww.liv.ac.uk/k1065489/ . ↑c.pp. 8, 13, and 15 [Far96] Farjoun, E. D. Cellular Spaces, Null Spaces and Homotopy Localization. Lecture Notes in Mathematics, vol. 1622. Springer, Berlin, Heidelberg, 1996. xiv+206 pp. DOI:10.1007/BFb0094429, MR1392221 (98f:55010). ↑c.p. 4 [GM03] Gel’fand, S. I., and Manin, Y. I. Methods of Homological Algebra, 2nd ed. Springer Monographs in Mathematics. Springer-Verlag, Berlin, Heidelberg, 2003. English, translated from Russian. xx+372 pp. DOI:10.1007/978-3-662-12492-5, MR1950475 (2003m:18001). ↑c.pp. 1, 9, and 10 [GZ67] Gabriel, P., and Zisman, M. Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, band 35. Springer, Berlin, Heidelberg, 1967. x+168 pp. DOI:10.1007/978-3-642-85844-4, MR0210125 (35 #1019). ↑c.pp. 1, 7, 8, and 10

[Hat02] Hatcher, A. Algebraic Topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. MR1867354 (2002k:55001). ↑c.p. 4 [KS06] Kashiwara, M., and Schapira, P. Categories and Sheaves. Grundlehren der mathematischen Wissenschaften, vol. 332. Springer, Berlin, Heidelberg, 2006. x+498 pp. DOI:10.1007/3-540-27950-4, MR2182076 (2006k:18001). ↑c.pp. 1, 7, and 14 [ML98] Mac Lane, S. Categories for the Working Mathematician, 2nd ed. Graduate Texts in Mathematics, vol. 5. springer, New York, NY, 1998. xii+317 pp. DOI:10.1007/978-1-4757-4721-8, MR1712872 (2001j:18001). ↑c.pp. 1, 2, 3, 4, 10, and 14 [MP12] May, J., and Ponto, K. More Concise Algebraic Topology: Localization, completion, and model categories. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2012. xxviii+514 pp. UCP:M/bo12322308.html, MR2884233 (2012k:55001). ↑c.p. 4 [Pos77] Postnikov, M. M. Localization of topological spaces. Russian Mathematical Surveys 32 (Dec 1977), no. 6, 121–184. DOI:10.1070/RM1977v032n06ABEH003860, MR0515419 (58 #24258). ↑c.p. 4 [PS14] Ponto, K., and Shulman, M. Traces in symmetric monoidal categories. Expositiones Mathematicae 32 (2014), no. 3, 248–273. DOI:10.1016/j.exmath.2013.12.003, MR3253568. ↑c.p. 14 [Sta16] Stacks Project Authors. Stacks Project, 2016. Accessed from: http://stacks.math.columbia.edu. ↑c.p. 9 [Sul05] Sullivan, D. Geometric Topology Localization, Periodicity, and Galois Symmetry: (the 1970 mit notes). K-Monographs in Mathematics, vol. 8. Springer, Dordrecht, Netherlands, 2005. xiv+283 pp. MR2162361 (2006m:55002). ↑c.pp. 1, 3, and 4 [Whi11] White, D. Bousfield localization of monoidal model categories. Talk at Graduate Student Seminar (Wesleyan University, September, 2011). 15 September version, accessed from: http://dwhite03.web.wesleyan.edu/files/LectureNotes/ BousfieldLocalization(GSS).pdf . ↑c.p. 1 Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, England, UK E-mail address: [email protected]