Re: module for a category

[rjwood@mathstat.dal.ca (RJ Wood)]

Here is another twist on this circle of ideas which appeared in the
introductory chapter of my 1976 thesis. Robin Cockett and I are
working on a redevelopment of it.

For monoidal (V,\ten, i), (promonoidal V will suffice) consider Brian Day's
convolution (closed) monoidal structure on set^{V^op}. If A is a
set^{V^op} category, it is helpful to think of A(-,-):A^op x A ---> set^{V^op}
as A(-,-,-):A^op x V^op x A ---> set with the interpretation that A(a,v,b)
provides a set of `v-indexed families' of arrows from a to b. The
composite of a v-indexed family  (v;f):a--->b with a w-indexed family
(w;g):b--->c is a w\ten v family (w\ten v;gf):a--->c. Of course it may
happen that for each a,b in A, A(a,-,b) is representable, by an object
A[a,b] in V. In this case each (v;f):a--->b takes the form
f:v--->A[a,b]. If for each a in A, and each v in V, A(a,v,-) is representable,
by an object a.v in A, then the (v;f):a--->b take the form
f:a.v--->b. Note that the identity a.v--->a.v considered as a v-indexed
family (v,j):a--->a.v can be construed as a family of `sum-injections'
for the `multiple' a.v. (Asking for a representing object {v,b} for
A(-,v,b) leads to dual considerations.) Simultaneous representability in
a,b and a,v is equivalent to the notion of `tensored V-category' mentioned
below.

In part this work was motivated by questions raised by Linton in `The
multilinear Yoneda lemmas' SLN 195, 209--229, and also pursued by
Reynolds in his 1973 Wesleyan thesis. For example, if A is a V-category
and M is a V-actegory, in the nomenclature below, what is a V-functor
A--->M, a V-functor M--->A? The familial approach, suggested by the
1970s work of Benabou, Pare/Schumacher, Rosebrugh and others provides
a straightforward intuitive answer. For general set^{V^op}-categories
A and M, the data for a set^{V^op}-functor F:A--->M sends, for each v in V,
each v-indexed family (v;f):a--->b to a v-indexed family (v;Ff):Fa--->Fb.
Each representability possibility for A and B allows for a compact presentation
of the data. When A is a V-category then it suffices to know F on
the generic families g:A[a,b]--->A[a,b]. In other words, one requires
(A[a,b];Fg):Fa--->Fb. If M is also a V-category then Fg is what is
usually denoted F_{a,b}:A[a,b]--->M[Fa,Fb], the effect of F on homs,
but if M is a V-actegory it will take the form Fa.A[a,b]--->Fb. If
A is a V-actegory then it suffices to know F on the generic
(v,j):a--->a.v. For M a V-category we have Fj:v--->M[Fa,F(a.v)], while
for M also a V-actegory we have Fj:Fa.v--->F(a.v), a form called `tensorial
strength' by Anders Kock in a seeries of papers about mononoidal monads.

In fact the 3x3 possibilities for `strengths' can be tabulated easily using
these considerations: Write 1) for `powers' {v,b}, 2) for homs [a,b] and
3) for `multiples' a.v. Then the i,j th entry below provides the form of
strength for a set^{V^op}-functor F:A--->M where A is of type i) and
M is of type j)

		 1)                     2)                  3)

    1)    F{v,b}--->{v,Fb}     v--->M[F{v,b},Fb]     F{v,b}.v--->Fb

    2)    Fa--->{A[a,b],Fb}    A[a,b]--->M[Fa,Fb]    Fa.A[a,b]--->Fb

    3)    Fa--->{v,F(a.v)}     v--->M[Fa,F(a.v)]     Fa.v--->F(a.v)

Best regards
RJ Wood

>  What you are looking for may be similar to something I queried Ross Street in regard to earlier this summer.
>  I'll save him some time by putting here the relevant part of his response.
>
>  > I think the one you first
>  >mention is what we have been calling V-actegories.  Benabou looked at
>  >these rather than (as well as?) V-categories in the early days of
>  >monoidal categories.  Pareigis also made use of them. More recently,
>  >publications of Paddy McCrudden involve them. There is a close
>  >connection with V-categories.  A V-module  V x A --> A  in this sense
>  >for which we have a parametrized adjoint  V(x,[a,b]) =~ A(x.a,b)
>  >makes  A  a V-category with V-valued hom [a,b].
>  >
>  >Conversely, a tensored V-category becomes such a V-module.
>
>  I recommend the work of McCrudden, who has developed among other things a
>  descent theoretic approach to the tensor product of V-actegories.
>  There is also resource in the work of Harald Lindner.
>  His paper, Enriched Categories and Enriched Modules, in Cahiers, Vol XXII-2 (1981)
>  develops morphisms between enriched categories and actegories, which he calls modules.
>  I'm curious about why it is that I have never seen his work referenced.
>
>  Paul B Levy writes:
>  >
>  > Hi
>  >
>  > Is there a standard reference for the notion of "left module for a
>  > category"?  (or right module, or bimodule)
>  >
>  > Is there any reference in the setting of ordinary categories rather than
>  > (or as well as) enriched categories or bicategories?
>  >
>  > Thanks
>  > Paul