Date: Sat, 31 Dec 1994 11:06:47 -0400 (AST)
Subject: composition of enriched functors

Date: Wed, 14 Dec 94 13:08:12 CST
From: David Yetter <dyetter@math.ksu.edu>


        It sticks in my mind that somewhere (probably Kelly's book
on enriched category theory) I have seen the composition of
enriched functors expressed in terms of coends.

        Our library's copy of Kelly is out, so I was hoping someone
could help me recover the result and all necessary hypotheses.

        More precisely, I recall that it ran like this (and have
proved this for ordinary categories):

        Associate to an enriched functor F:A \rightarrow B
the functor Hom(F(-),-):A^{op} \times B \rightarrow V

        The composition of F and G is then given by

        \integral^c Hom(F(-),c) \otimes Hom(G(c),-) \cong Hom(G(F(-)),-).

        I am particularly interested in an analogous result for
C-linear categories, and categories enriched in the category of
C-linear categories, as this could have applications to topological
and conformal field theories.

Best Thoughts to all,
David Yetter

Date: Tue, 3 Jan 1995 15:08:25 -0400 (AST)
Subject: Re: composition of enriched functors

Date: Mon, 2 Jan 95 09:34:16 +1100
Subject: Re: composition of enriched functors

David Yetter, in his query, seems to be making heavy weather of the compositionof enriched functors. The composite of F:A --> B and G:B --> C is given on
objects by GFa, while its "effect on maps" is the composite of the effects of Fand of G as in  A(a,a') ------> B(Fa,Fa') -----> C(GFa,GFa'). That is all.

 There is further a composition of profunctors (also called bimodules, or just
modules), given by an evident co-end formula. Now every functor F determines a
profunctor F*, where F*(a,b) is B(Fa,b); and the calculation David refers to
(which is a simple application of the Yoneda isomorphism) is just the verification
that (GF)* is canonically isomorphic to (G*)(F*). Of course this holds, in
particular, for C-linear categories.

 New-Year greetings to all - Max Kelly.

Date: Tue, 3 Jan 1995 15:11:26 -0400 (AST)
Subject: David Yetter's question

Date: Tue, 3 Jan 1995 14:43:10 +1100
From: Ross Street <street@macadam.mpce.mq.edu.au>

>The composition of F and G is then given by
>
>        \integral^c Hom(F(-),c) \otimes Hom(G(c),-) \cong Hom(G(F(-)),-).

This follows from the enriched Yoneda Lemma. The setting is as follows.
It is tensor product of (bi)modules which is given by the coend formula.
Modules (= profunctors = distributors) between V-categories are the arrows
for a bicategory Mod(V) with this tensor product as composition. Each
V-functor  F  becomes a module in two ways: F_* and F^*. The latter
involves the Hom(F(a),c) as required by David. Each process gives a
locally fully faithful embedding of V-Cat in V-Mod (Yoneda's Lemma)
with appropriate variances. Moreover,  F^*  is right adjoint to F_* in
the bicategory Mod(V).

A good reference is Lawvere's paper "Metric spaces, generalised logic,
and closed categories" Rend. Sem. Mat. Fis. Milano 43 (1974) 135-166.

Best wishes for the New Year,
Ross


Date: Mon, 9 Jan 1995 08:22:03 -0400 (AST)
Subject: Re: composition of enriched functors

Date: Mon, 9 Jan 1995 09:02:15 +0100
From: Axel Poigne <poigne@gmd.de>

Referring to recent contibutions by Yetter and Kelly.

As remarked by Kelly, the construction is straightforward. A paper
strongly recommended to be read in this context is Lawvere's 'Generalized
Metric Spaces ...". The material about composition of profunctors +
construction of "free algebras" are to be found in my contribution "Basic
category theory", chpt. 6, in the Handbook of Logic in Computer Science
(Abramsky, Gabbay, Maibaum, eds.).


Axel Poigne