Finn Lawler


If KK and LL are bicategories, then a biprofunctor H:KLH \colon K ⇸ L is a pseudofunctor H¯:L op×KCat\bar H \colon L^{op} \times K \to Cat.

To define the tricategory BiProfBiProf of biprofunctors, we need to know that PK=[K op,Cat]P K = [K^{op}, Cat] is the free 2-cocompletion of KK. Then BiProfBiProf can be defined as having objects bicategories K,L,K,L,\ldots and hom-bicategories hom(K,L)\hom(K,L) the strict 2-categories of cocontinuous pseudofunctors PKPLP K \to P L.

Given H:PKPLH \colon P K \to P L and G:PLPMG \colon P L \to P M, their composite GHG H corresponds to the pseudofunctor GH¯(m,k)=(GHyk)m=H¯(,k)G¯(m,)\bar{G H}(m,k) = (G H y k) m = \bar H(-, k) \star \bar G(m,-), the colimit of G¯(m,)\bar G(m,-) weighted by H¯(,k)\bar H(-,k). Using the bicategorical co-Yoneda lemma and a couple of other tricks from Kelly section 3.3, we can write this as

GH¯(m,k)=hom L(H¯(,k)×G¯(m,)) \bar{G H}(m,k) = \hom_L \star (\bar H(-,k) \times \bar G(m,-))

showing that the composite GH¯\bar{G H} of profunctors is indeed a ‘coend’ H(,)×G(,)\int^\ell H(\ell,-) \times G(-,\ell).

The co-Yoneda lemma then shows that if F:KLF \colon K \to L and G:JMG \colon J \to M are functors, and H:LMH \colon L ⇸ M is a profunctor, then

M(G,1)HL(1,F)H(G,F) M(G,1) \circ H \circ L(1,F) \simeq H(G,F)

If L(1,F)L(1,F) is taken as a functor KPLK \to P L, then the corresponding cocontinuous functor is its left Kan extension along the Yoneda embedding, which by the usual nerve and realization business has a right adjoint given by the pullback-along-FF functor F *:PLPK:V,kVFkF^* \colon P L \to P K \colon V, k \mapsto V F k. By the co-Yoneda lemma this latter is VL(Fk,)V \star L(F k, -), so that the right adjoint of L(1,F)¯\bar{L(1,F)} is equivalently L(F,1)¯\bar{L(F,1)}. Hence in BiProfBiProf

L(1,F)L(F,1) L(1,F) \dashv L(F,1)

as in ProfProf.

Kleisli objects

Suppose T:KKT \colon K ⇸ K is a pseudomonad in BiProfBiProf. Its Kleisli object is the bicategory K TK_T with objects those of KK and hom-categories K T(k,)=T(k,)K_T(k,\ell) = T(k,\ell), with composition defined using the multiplication of TT. The unit of TT supplies a functor F T:KK TF_T \colon K \to K_T that is the identity on objects, and clearly TK T(F T,F T)T \simeq K_T(F_T, F_T).

The statement that K TK_T is the Kleisli object of TT is the statement that K T(1,F T)K_T(1,F_T) is the universal right TT-module (or right K T(F T,F T)K_T(F_T,F_T)-module).

We have as before the adjunction K T(1,F T)K T(F T,1)K_T(1,F_T) \dashv K_T(F_T,1), and precomposition gives an adjunction between BiProf(K,L)BiProf(K, L) and BiProf(K T,L)BiProf(K_T, L), with left adjoint HHK T(F T,1)H \mapsto H \circ K_T(F_T,1). Then RMod(T,L)RMod(T,L) is equivalent to the category of algebras for the monad induced by this adjunction, so there is a comparison functor BiProf(K T,L)RMod(T,L)BiProf(K_T,L) \to RMod(T,L), given by composition with the right TT-module (K T(1,F T),ϵK T(1,F T))(K_T(1,F_T), \epsilon \circ K_T(1,F_T)). This comparison functor is an equivalence if the right adjoint U:GGK T(1,F T)U \colon G \mapsto G \circ K_T(1,F_T) is monadic in the sense of LMV, that is if it preserves UU-split codescent objects and reflects adjoint equivalences. BiProfBiProf has local colimits, which because composition is given by coends are stable under composition on both sides, so BiProf(K T,L)BiProf(K_T,L) has, and UU preserves, the required codescent objects. Suppose α:GH\alpha \colon G \Rightarrow H is a transformation; then because F TF_T is the identity on objects, the components of α\alpha are exactly the components of αK T(1,F T)\alpha \circ K_T(1,F_T), and so if the latter are all equivalences then so are the former. Hence U=K T(1,F T)U = - \circ K_T(1,F_T) is monadic, and BiProf(K T,L)RMod(T,L)BiProf(K_T,L) \sim RMod(T,L).

Revised on March 1, 2012 07:19:56 by Finn Lawler? (