Colimits and Profunctors Robert Par´e Dalhousie University [email protected] June 14, 2012 The Problem For two diagrams I JJ Γ Φ A what is the most general kind of morphism Γ / Φ which will produce a morphism lim Γ lim Φ? −! / −! Answer: A morphism lim Γ lim Φ. −! / −! Want something more syntactic? E.g. F IIJ/ J Γ Φ A The Problem For two diagrams I JJ Γ Φ A what is the most general kind of morphism Γ / Φ which will produce a morphism lim Γ lim Φ? −! / −! Answer: A morphism lim Γ lim Φ. −! / −! Want something more syntactic? E.g. F IIJφ / J +3 Γ Φ A Example A0 B0 g0 g1 f f 0 1 A / B B 1 p1 1 2 f h k A B p1f0 = g0p2 p1f1 = g0p3 g1p2 = g1p3 Thus we get hp1f0 = hg0p2 = kg1p2 = kg1p3 = hg0p3 = hp1f1 So there is a unique p such that pf = hp1. Problems I Different schemes (number of arrows, placement, equations) may give the same p I It might be difficult to compose such schemes On the positive side I It is equational so for any functor F : A / B for which the coequalizer and pushout below exist we get an induced morphism q The Problem (Refined) For two diagrams I JJ Γ Φ A what is the most general kind of morphism Γ / Φ which will produce a morphism lim F Γ lim F Φ −! / −! for every F : A B for which the lim's exist? / −! I Should be natural in F (in a way to be specified) op Take F to be the Yoneda embedding Y : A / SetA . Then we have the bijections lim Y Γ lim Y Φ −! / −! lim A(−; ΓI ) / lim A(−; ΦJ) −!I −!J hA(−; ΓI ) / lim A(−; ΦJ)i −!J I hx 2 lim A(ΓI ; ΦJ)i I −!J I An element of lim A(ΓI ; ΦJ) is an equivalence class of morphisms −!J a [ΓI / ΦJ]J where a ∼ a0 iff there is a path of diagrams ak ΓI / ΦJk Φjk ΓI / ΦJk+1 ak+1 Thus, to give a compatible family hx 2 lim A(ΓI ; ΦJ)i I −!J I we must give: aI I For each I , a JI and a morphism ΓI / ΦJI 0 i I For each I / I a path of J's and a's joining 0 Γi aI ΓI / ΓI / ΦJI with 0 ΓI / ΦJI 0 aI 0 0 0 Two such choices haI :ΓI / ΦJI i are haI :ΓI / ΦJI i are aI equivalent, if for each J there is a path joining ΓI / ΦJI with a0 I 0 ΓI / ΦJI . Theorem The above data induces for every F a morphism lim F Γ lim F Φ. Two such sets of data induce the same −! / −! morphism for all F iff they are equivalent as described above. p2 A0 / B0 g0 g1 f f 0 1 A / B B 1 p1 1 2 f1 p1 A0 / A1 / B1 O g0 f0 p1 p3 A0 / A1 / B1 A0 / B0 O g0 g1 A0 / B0 A0 / B2 p2 O g1 A / B 0 p2 0 Canonization Recalling our idea of F IIJ/ J φ +3 Γ Φ A we get for every I , a JI = FI , and a morphism aI = φI :ΓI / ΦFI . Naturality of φ gives a one-step path Γi φI ΓI 0 / ΓI / ΦFI ΦFi ΓI 0 / ΦFI 0 φI 0 In the general case I JI is not a functor. There can be several 0 JI , and for i : I / I we don't get a morphism JI / JI 0 but only a path. This is a kind f \relation between categories". They are called profunctors (distributors, bimodules, modules, relators). Profunctors op I A profunctor P : A • / B is a functor P : A × B / Set I Every functor F : A / B gives two profunctors op F∗ : A • / B; F∗ = B(F −; −): A × B / Set F ∗ : B • / A; F ∗ = B(−; F −): Bop × A / Set ∗ F∗ a F P Q I Composition A • / B • / C Z B Q ⊗ P(A; C) = Q(B; C) × P(A; B) x y = f[A • / B • / C]B g = fy ⊗B xg P Q 0 x y x0 0 y I A • / B • / C ∼ A • / B • / C if there is x y AAB• / BBC• / C y ⊗ x = y 0b ⊗ x b = y 0 ⊗ bx AAB• / B0 • / C = y 0 ⊗ x0 x0 y 0 Given functors Γ Φ I / A o J ∗ we get a profunctor Φ ⊗ Γ∗ : I • / J ∗ Φ ⊗ Γ∗(I ; J) = A(ΓI ; ΦJ): Proposition A compatible family hx 2 lim A(ΓI ; ΦJ)i determines a I J J ∗ −! profunctor P ⊆ Φ ⊗ Γ∗ with the property that for every F and every a 2 P(I ; J) we have Fa F ΓIIF/ F ΦJ injI injJ lim F Γ lim F Φ −! / −! for the morphism induced by hxi i. Proof. P(I ; J) = fa :ΓI / ΦJj[a] = [xI ]g. Total Profunctors Definition P : A • / B is total if for every A, lim P(A; B) =∼ 1: −!B Let T : A / 1 be the unique functor. Then P is total iff =∼ T∗ ⊗ P / T∗. Proposition (1) Total profunctors are closed under composition. (2) For any functor F : A / B,F∗ is total. (In particular IdA is 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. Σ Θ ∗ (6) For I o K / J, Θ∗ ⊗ Σ is total iff Σ is final. Profunctors over A Definition For Γ : I / A and Φ : J / A, a profunctor from Γ to Φ (or a profunctor from I to J over A) is P IIJ• / J π Γ Φ A where P is a profunctor I • / J and ∗ π : P / A(Γ−; Φ−) = Φ ⊗ Γ∗ is a natural transformation. Profunctors over A compose in the \obvious" way: (Q; ) ⊗ (P; π) = (Q ⊗ P; ⊗ π) ⊗ π(y ⊗ x) = ( y)(πx): Theorem Let P IIJ• / J π Γ Φ A be a profunctor over A with P total. Then for every F : A / B for which lim F Γ and lim F Φ exist, there is a unique morphism −! −! lim F π : lim F Γ lim F Φ such that for every x 2 P(I ; J) we have −! −! / −! F π(x) F ΓIIF/ F ΓJ injI injJ lim F Γ/ lim F Φ −! lim F φ −! −! If (Q; ):Φ / Ψ is another total profunctor over A, we have lim F ( ⊗ π) = (lim F )(lim F π): −! −! −! Saturation Definition P / / Q : I • / J is saturated if x 2 Q(I ; J) and for some j j : J / J0, jx 2 P(I ; J0) implies x 2 P(I ; J). I P is saturated in Q iff for every I , P(I ; −) / / Q(I ; −) is complemented in SetJ. I Every P / / Q has a saturation P¯ / / Q. Theorem Let (P; π) and (P0; π0) be two total profunctors Γ • / Φ. Then they induce the same family lim F Γ / lim F Φ iff the images of ∗ 0 0 −! ∗ −! π : P / Φ ⊗ Γ∗ and π : P / Φ ⊗ Γ∗ have the same saturation. Naturality Definition A family of morphisms b : lim F Γ lim F Φ is natural if for F −! / −! every G we have b lim GF Γ GF lim GF Φ −! / −! G lim F Γ / G lim F Φ −! GbF −! 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 corresponds to a lax normal functor I / Prof where an object I is 0 sent to KI , the fibre over I and a morphism i : I / I to the profunctor Pi : KI • / KI 0 given by formula 0 k 0 Pi (K; K ) = fK / K jΛk = ig Definition Λ: K / I is a cohesive family of categories if each Pi is total. In elementary terms, for every K in K and every morphism k i :ΛK / I 0, there exists a morphism k : K / K 0 such that i = Λk and any two such liftings are connected by a path over i. k KKK/ K 0 ΛKIK / I 0 i Proposition (1) Opfibrations are cohesive families (2) Cohesive families are stable under pullback (3) Cohesive families are closed under composition Definition A cohesive family of diagrams in A is a span Γ KKA/ A Λ I with Λ cohesive. Let ΓI = ΓjKI . Theorem lim Γ extends to a unique functor lim Γ : I A such that for all −! I −! () / k : K / K 0 over i : I / I 0 Γk ΓK / ΓK 0 injK injK0 lim ΓI / lim Γ 0 −! lim Γ −! I −! i Kan Extensions lim Γ : I A is the left Kan extension and cohesiveness says it −! () / is fibrewise. So perhaps a more functorial version of the theorem is: Theorem Λ: K / I is cohesive iff for every pullback diagram F LLK/ K Σ Λ JIJ / I F and every cocomplete A, the canonical morphism F ∗ AL o AK Lan Lan Σ λ Λ AJ o AI F ∗ is an isomorphism. If we take = 1, F I 2 I, we get (Lan Γ)I ∼ lim Γ . J Λ = −! I The Comprehensive Factorization Set Cat Relations $ Profunctors Everywhere Defined $ Total Single Valued $ ? Functions $ Functors Recall the comprehensive factorization on Cat (Street & Walters '79).