Finn Lawler 2-end

I think the easiest way to define a bicategorical notion of end is to follow the identification in enriched category theory of ends with hom\hom-weighted limits.

The Yoneda lemma for 2-extranatural transformations shows that for a biprofunctor H:K op×KCatH \colon K^{op} \times K \to Cat there is an equivalence Exnat(1,H){hom K,H}Exnat(\mathbf{1}, H) \simeq \{\hom_K, H\}. For an arbitrary T:K op×KLT \colon K^{op} \times K \to L, we then find that Exnat(,T)L(,{hom K,T})Exnat(\ell, T) \simeq L(\ell, \{\hom_K, T\}) if that limit exists, so that we may write this limit as aT(a,a)\int_a T(a,a).

One fact will be useful, and it is easy to show: for pseudofunctors F,G:KLF, G \colon K \to L, the category Nat(F,G)=[K,L](F,G)Nat(F,G) = [K,L](F,G) is equivalent to aL(Fa,Ga)\int_a L(F a, G a) as usual in enriched category theory. This is easiest to see by comparing Nat(F,G)Nat(F,G) with Cat(1, aL(Fa,Ga))Exnat(1,L(F,G))Cat(\mathbf{1}, \int_a L(F a, G a)) \simeq Exnat(\mathbf{1}, L(F-, G-)).

Created on May 27, 2011 at 00:12:31. See the history of this page for a list of all contributions to it.