Finn Lawler biprofunctor (Rev #1)

Definition

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).

Revision on May 21, 2011 at 19:15:05 by Finn Lawler?. See the history of this page for a list of all contributions to it.