Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Triads over Quantaloids ˇ Radek Slesinger [email protected] Department of Mathematics and Statistics Masaryk University Brno

50th Summer School on Algebra and Ordered Sets Nov´y Smokovec, 4th September 2012 Supported by

Quantaloid triads

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Sup-lattice – a complete join-semilattice L W W Homomorphism f ( xi ) = f (xi ) Quantale – a sup-lattice binary operation W Q with W associativeW W satisfying q ( ri ) = (qri ) and ( ri ) q = (ri q) Unital if Q has a multiplicative unit e W W Homomorphism f ( qi ) = f (qi ) and f (qr ) = f (q)f (r ) Analogy: abelian groups, rings

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Examples Ideals of a ring (ideals generated by unions, multiplication of ideals) Binary relations on a set (unions of relations, composition of relations) Endomorphisms of a sup-lattice (pointwise suprema, composition of maps) Frames (open subsets of a topological space) ∧ as the binary operation Powerset of a semigroup (A · B = {a · b | a ∈ A, b ∈ B})

Quantaloid triads

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Right Q-module – a sup-lattice MWwith rightWaction of the quantale satisfying ( mi ) q = (mi q), W W m ( qi ) = (mqi ), m(qr ) = (mr )q Unital if Q is unital and me = m for all m W W Homomorphism f ( mi ) = f (mi ), f (mq) = f (m)q A sub-sup-lattice of a quantale closed under right multiplication by quantale elements A sup-lattice with action of the quantale of its endomorphisms f · m = f (m) (a left module) Left-sided elements of a quantale (s.t. q · l ≤ l for all q ∈ Q ⇐⇒ >Q · l ≤ l)

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Quantaloid – categorification of the concept of quantale – (small) category Q with Q(A, B) ∈ SupLat for any A, B ∈ Ob(Q) where composition of arrows distributes over arbitrary joins on both sides unital quantale – a one-object quantaloid SupLat, Quant, Q-Mod – large quantaloids Homomorphism of quantaloids – a SupLat-enriched functor F : Q → P (any Q(A, B) → P(FA, FB) is a sup-lattice homomorphism)

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

For convenience, we shall compose arrows in the “geometrical” way: in X a / Y b / Z we will use a.b for b ◦ a Left module over a quantaloid – a contravariant functor M : Qop → SupLat with action of Q for x ∈ MX and a : X → Y : a · x = (Ma)(x) (view: a collection of sup-lattices preserving the structure of Q) Right module – a covariant functor M : R → SupLat with action x · a = (Ma)(x)

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

(Q, R)-bimodule – a bifunctor M : Qop × R → SupLat where the actions commute: M(Q 0 , R) 8

M(q,R)

M(q,r )

M(Q, R) M(Q,r )

M(Q 0 ,r )

&

& / M(Q 0 R 0 ) 8 M(q,R 0 )

M(Q, R 0 ) Homomorphism of (bi)modules – a (bi)natural transformation between functors M and N, each fX : Mx → Nx is a sup-lattice homomorphism

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Q as a (Q, Q)-bimodule – take Q(−, −) ”bilinear form” – a binatural transformation from a (Q, Q)-bimodule M to Q(−, −)

Quantaloid triads

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Quantum triad (D. Kruml, 2008) (L, T , R) such that Quantale T Left T -module L Right T -module R (T , T )-bimorphism (homomorphism of respective modules when fixing one component) L × R → T , satisfying associativities TLR, LRT

T

< L×R b  ET T

Not specific for quantales – can be defined for monoids, rings, po-monoids, . . . as well.

T

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Example 1 L = right-sided elements of a quantale Q (r · >Q ≤ r ) R = right-sided elements of Q T = two-sided elements of Q Example 2 Sup-lattice 2-forms (P. Resende 2004) (∼ Galois connections) L, R sup-lattices T = 2 (the 2-element frame)

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Solution of the triad Quantale Q such that L is a (T , Q)-bimodule R is a (Q, T )-bimodule there is a (Q, Q)-bimorphism R × L → Q satisfying associativities QRL, RLQ, RTL, LQR, LRL, RLR

Quantum Triads

Q

< R ×L b  ET Q

Example of right/left/two-sided elements: Q is a solution

Quantaloid triads

Q

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Two special solutions Q0 = R ⊗T L (r1 ⊗ l1 ) · (r2 ⊗ l2 ) = r1 (l1 r2 )⊗ l2 l 0 (r ⊗ l) = (l 0 r )l (r ⊗ l)r 0 = r (lr 0 ) Q1 = {(α, β) ∈ End(L) × End(R) | α(l)r = lβ(r ) for all l ∈ L, r ∈ R} (α1 , β1 ) · (α2 , β2 ) = (α2 ◦ α1 , β1 ◦ β2 ) l 0 (α, β) = α(l 0 ) (α, β)r 0 = β(r 0 )

Quantaloid triads

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Couple of solutions There is a φ : Q0 → Q1 , φ(r ⊗ l) = ((− · r )l, r (l · −)) which forms a so-called couple of quantales (Egger, Kruml 2008): Q0 is a (Q1 , Q1 )-bimodule with φ(q)r = qr = qφ(r ) for all q, r ∈ Q0 All solutions Q of (L, T , R) then provide factorizations of the couple: There are maps φ0 : Q0 → Q and φ1 : Q → Q1 s.t. φ1 ◦ φ0 = φ φ0 (φ1 (k)q) = kφ0 (q) and φo (qφ1 (k)) = φ0 (q)k (so φ0 becomes a coupling map under scalar restriction along φ1 )

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Example L is a sup-lattice, R = T = 2 L × 2 → 2 : (0, y ) 7→ 0, (x, 0) 7→ 0, (x, 1) 7→ 1 Then

( y Q0 = 2 ⊗2 L = L with xy = 0

x= 6 0 x =0

Q1 = {(x 7→ 0, y 7→ 0), (idL , idR )} = 2

Quantum Triads

Quantaloid triads

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Triad over a quantaloid (L, T , R) such that Quantaloid T Left T -module L Right T -module R (T , T )-bimorphism L × R → T , satisfying associativities T LR, LRT

T

< L×R b  ET T

T

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Example 1 Q = a quantaloid L = right-sided arrows (l : X → Y s.t. l ≤ l.>Y ) of Q R = left-sided (r : X → Y s.t. r ≤ >x .r ) arrows of Q T = two-sided (both right- and left-sided) elements of Q Example 2 T = a quantaloid fix an object X of T L = T (−, X ) R = T (X , −)

Quantaloid triads

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Solution of the triad Quantaloid Q such that L is a (T , Q)-bimodule R is a (Q, T )-bimodule there is a (Q, Q)-bimorphism R × L → Q satisfying associativities QRL, RLQ, RT L, LQR, LRL, RLR

Q

< R×L b  ET Q

More precisely, there are bifunctors L0 , R0 that restrict to L, R when fixing the T -argument. The sup-lattices of the T -module L form a subcategory of the (T , Q)-bimodule L0 . An analogy holds for R with R0 .

Q

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Example 1 Q = a quantaloid L = right-sided arrows (l : X → Y s.t. l ≤ l.>Y ) of Q R = left-sided arrows (r : X → Y s.t. r ≤ >x .r ) of Q T = two-sided (both right- and left-sided) elements of Q Q is a solution Example 2 T = a quantaloid fix an object X of T L = T (−, X ) R = T (X , −) T (X , X ) is a solution

Quantaloid triads

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

Goal Provide characterization of the category of solutions, analogous to the “one-object” setting. Possibly with Q0 = M ⊗T N – a full subquantaloid of SupLat with objects MX ⊗NY for X , Y in T Q1 = (bi)modules as objects, (bi)natural transformations as arrows and some coupling between Q0 and Q1

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Quantaloid triads

References J. Egger, D. Kruml: Girard couples of quantales, Applied categorical structures, 18 (2008), pp. 123–133 D. Kruml: Quantum triads: an algebraic approach, http://arxiv.org/abs/0801.0504 P. Resende: Sup-lattice 2-forms and quantales, Journal of Algebra, 276 (2004), pp. 143167 K. I. Rosenthal: The theory of quantaloids, Pitman Research Notes in Mathematics Series 348, Longman, 1996

Sup-lattices/Quantales/Quantale Modules

Quantaloids/Quantaloid Modules

Quantum Triads

Thank you for your attention.

Quantaloid triads