Colimits and Profunctors Robert Par e Dalhousie University - PowerPoint PPT Presentation

Colimits and Profunctors Robert Par´ e Dalhousie University pare@mathstat.dal.ca June 14, 2012

� � � � The Problem For two diagrams I I J J Γ Φ A A what is the most general kind of morphism Γ � Φ which will produce a morphism � lim lim → Γ → Φ ? − − � lim Trivial answer: A morphism lim → Γ → Φ. − − We want something more syntactic! E.g. F � J I I J Γ Φ A A

� � � � The Problem For two diagrams I I J J Γ Φ A A what is the most general kind of morphism Γ � Φ which will produce a morphism � lim lim → Γ → Φ ? − − � lim Trivial answer: A morphism lim → Γ → Φ. − − We want something more syntactic! E.g. F � J I I J φ � Γ Φ A A

� � � � � � � Example p 3 A 0 A 0 B 0 B 0 p 1 f 0 = g 0 p 2 p 2 g 0 g 1 f 0 � f 1 p 1 f 1 = g 0 p 3 A 1 A 1 A 1 p 1 � B 1 B 1 B 1 B 1 B 2 B 2 B 2 B 2 g 1 p 2 = g 1 p 3 f �� h k A B B Then we get hp 1 f 0 = hg 0 p 2 = kg 1 p 2 = kg 1 p 3 = hg 0 p 3 = hp 1 f 1

� � � � � � � � Example p 3 A 0 A 0 B 0 B 0 p 1 f 0 = g 0 p 2 p 2 g 0 g 1 f 0 � f 1 p 1 f 1 = g 0 p 3 A 1 A 1 A 1 p 1 � B 1 B 1 B 1 B 1 B 2 B 2 B 2 B 2 g 1 p 2 = g 1 p 3 f �� h k A B B p Thus we get hp 1 f 0 = hg 0 p 2 = kg 1 p 2 = kg 1 p 3 = hg 0 p 3 = hp 1 f 1 So there is a unique p such that pf = hp 1 .

� � � � � � � � � Problems ◮ Different schemes (number of arrows, placement, equations) may give the same p ◮ It might be difficult to compose such schemes On the positive side � B for which the ◮ It is equational so for any functor F : A coequalizer and pushout below exist we get an induced morphism q Fp 3 FA 0 FA 0 FB 0 FB 0 Fp 2 Fg 0 Fg 1 Ff 0 � Ff 1 FA 1 FA 1 FA 1 FB 1 FB 1 FB 1 FB 1 FB 2 FB 2 FB 2 FB 2 Fp 1 f �� h k C D D q

� � The Problem (Refined) For two diagrams in A I I J J Γ Φ A A what is the most general kind of morphism Γ � Φ which will produce a morphism � lim lim → F Γ → F Φ − − � B for which the lim for every F : A → ’s exist? − ◮ It should be natural in F (in a way to be specified)

� First Solution � Set A op . Then we Take F to be the Yoneda embedding Y : A have the bijections � lim lim → Y Γ → Y Φ − − � lim lim → I A ( − , Γ I ) → J A ( − , Φ J ) − − � lim � A ( − , Γ I ) → J A ( − , Φ J ) � I − � x I ∈ lim → J A (Γ I , Φ J ) � I − An element of lim → J A (Γ I , Φ J ) is an equivalence class of morphisms − a � Φ J ] J [Γ I where a ∼ a ′ iff there is a path of diagrams a k � Φ J k Γ I Γ I Φ J k Φ j k Γ I Γ I a k +1 � Φ J k +1 Φ J k +1 joining a to a ′ .

� Theorem Suppose we are given a I � Φ J I ◮ For each I, a J I and a morphism Γ I i � I a path of J’s and a’s joining ◮ For each I ′ a I Γ i � Γ I � Φ J I Γ I ′ to Γ I ′ Φ J I ′ a I ′ � lim then for every F we get a morphism lim → F Γ → F Φ . Two such − − � Φ J I � and � a ′ � Φ J ′ choices, � a I : Γ I I : Γ I I � , induce the same � lim morphisms lim → F Γ → F Φ , iff for each J there is a path joining − − a ′ a I � Φ J I to Γ I I � Φ J ′ Γ I I .

� � � � � � � � � � � Example Again p 2 A 0 A 0 B 0 B 0 g 0 g 1 f 0 � f 1 A 1 A 1 p 1 � B 1 B 1 B 2 B 2 f 1 � A 1 p 1 � B 1 A 0 A 0 A 1 A 1 A 1 B 1 g 0 f 0 � A 1 p 1 � B 1 p 3 � B 0 A 0 A 0 A 1 A 1 A 1 B 1 A 0 A 0 A 0 A 0 B 0 B 0 B 0 g 0 g 1 � B 2 A 0 A 0 A 0 B 0 B 0 B 0 A 0 A 0 A 0 A 0 B 2 B 2 B 2 p 2 g 1 A 0 A 0 B 0 B 0 p 2

� � � � Canonization Recalling our first idea of F � J I I J φ � Γ Φ A A where we get for every I , a J I = FI , and a morphism � Φ FI . Naturality of φ gives a one-step path a I = φ I : Γ I φ I � Φ FI Γ i � Γ I Γ I ′ Γ I ′ Γ I Γ I Γ I Φ FI Φ Fi Γ I ′ Γ I ′ Γ I ′ Φ FI ′ Φ FI ′ Φ FI ′ φ I ′ In the general case I � J I is not a functor. There can be several � I we don’t get a morphism J I ′ � J I but only J I , and for i : I ′ a path. This is a kind of “relation between categories”. They are called profunctors (distributors, bimodules, modules, relators).

� � � � Profunctors � B is a functor P : A op × B � Set ◮ A profunctor P : A • � B gives two profunctors ◮ Every functor F : A F ∗ = B ( F − , − ) : A op × B � B , � Set F ∗ : A • F ∗ : B F ∗ = B ( − , F − ) : B op × A � A , � Set • F ∗ ⊣ F ∗ Q P � B � C ◮ Composition A • • � B Q ⊗ P ( A , C ) = Q ( B , C ) × P ( A , B ) y x � B � C ] B } = { y ⊗ B x } = { [ A • • P Q y ′ y x x ′ � B � C ∼ A � B ′ � C if there is ◮ A • • • • y x � B � C A A B B B C y ⊗ x = y ′ b ⊗ x • • = y ′ ⊗ bx b = y ′ ⊗ x ′ A A B ′ B ′ B ′ B ′ C C • • x ′ y ′

� � For example, given functors Γ Φ � A � I J we get an easily computed profunctor Φ ∗ ⊗ Γ ∗ : I � J • Φ ∗ ⊗ Γ ∗ ( I , J ) = A (Γ I , Φ J ) . Proposition A compatible family � x I ∈ lim → J A (Γ I , Φ J ) � J determines a − subprofunctor P ⊆ Φ ∗ ⊗ Γ ∗ with the property that for every F and every a ∈ P ( I , J ) we have Fa � F Φ J F Γ I F Γ I F Φ J inj I � inj J lim lim → F Γ → F Γ lim lim → F Φ → F Φ − − − − for the morphism induced by � x I � . Proof. � Φ J | [ a ] = [ x I ] } . P ( I , J ) = { a : Γ I

Total Profunctors Definition � B is total if for every A , P : A • → B P ( A , B ) ∼ lim = 1 . − � 1 be the unique functor. Then P is total iff Let T : A ∼ = � T ∗ . T ∗ ⊗ P Proposition (1) Total profunctors are closed under composition. � B , F ∗ is total. (In particular Id A is (2) For any functor F : A total.) (3) If P and P ⊗ Q are total then Q is total. (4) Total profunctors are closed under connected colimits and quotients. (5) F ∗ is total iff F is final. � J , Θ ∗ ⊗ Σ ∗ is total iff Σ is final. Σ Θ (6) For I � K

� � Profunctors over A Definition � A and Φ : J � A , a profunctor from Γ to Φ (or a For Γ : I profunctor from I to J over A ) is P � J I I J • � π Γ Φ A A � J and where P is a profunctor I • � A (Γ − , Φ − ) = Φ ∗ ⊗ Γ ∗ is a natural transformation. π : P Profunctors over A compose in the “obvious” way: ( Q , ψ ) ⊗ ( P , π ) = ( Q ⊗ P , ψ ⊗ π ) ψ ⊗ π ( y ⊗ x ) = ( ψ y )( π x ) .

� � � � Theorem Let P � J I I J • � π Γ Φ A A � B be a profunctor over A with P total. Then for every F : A for which lim → F Γ and lim → F Φ exist, there is a unique morphism − − � lim lim → F π : lim → F Γ → F Φ such that for every x ∈ P ( I , J ) we have − − − F π ( x ) � F Φ J F Γ I F Γ I F Φ J inj I � inj J lim lim → F Γ → F Γ lim lim → F Φ → F Φ − − − − → F φ lim − � Ψ is another total profunctor over A , we have If ( Q , ψ ) : Φ lim → F ( ψ ⊗ π ) = (lim → F ψ )(lim → F π ) . − − −

� � � � Saturation Definition � Q : I � J is saturated if x ∈ Q ( I , J ) and for some j P • � J ′ , jx ∈ P ( I , J ′ ) implies x ∈ P ( I , J ). j : J � Q ( I , − ) is ◮ P is saturated in Q iff for every I , P ( I , − ) complemented in Set J . � Q has a saturation ¯ � Q . ◮ Every P P Theorem � Φ . Then Let ( P , π ) and ( P ′ , π ′ ) be two total profunctors Γ • � lim they induce the same family lim → F Γ → F Φ iff the images of − − � Φ ∗ ⊗ Γ ∗ and π ′ : P ′ � Φ ∗ ⊗ Γ ∗ have the same saturation. π : P

� � � Naturality Definition � lim A family of morphisms b F : lim → F Γ → F Φ is natural if for − − every G we have b GF � lim lim lim → GF Γ → GF Γ lim → GF Φ → GF Φ − − − − G lim G lim → F Γ → F Γ G lim G lim → F Φ → F Φ − − − − Gb F Theorem A total profunctor over A induces a natural family as above. Every natural family comes from a total saturated profunctor ⊆ Φ ∗ ⊗ Γ ∗ . In fact there is a bijection between natural families and saturated total ⊆ Φ ∗ ⊗ Γ ∗ .

Cohesive Families As remarked by B´ enabou already in the 70’s, a category over I K Λ � I � Prof where an object I is corresponds to a lax normal functor I � I ′ to the sent to K I , the fibre over I , and a morphism i : I � K I ′ given by the formula profunctor P i : K I • k � K ′ | Λ k = i } P i ( K , K ′ ) = { K Definition � I is a cohesive family of categories if each P i is total. Λ : K

� Cohesive Families (Continued) In elementary terms, for every K in K and every morphism � K ′ such that � I ′ , there exists a morphism k : K k i : Λ K i = Λ k and any two such liftings are connected by a path over i . k � K ′ K ′ K K Λ K Λ K I ′ I ′ i Proposition (1) Opfibrations are cohesive families (2) Cohesive families are stable under pullback (3) Cohesive families are closed under composition