A remark related to Paul Levy's email on modules (Pat Donaly)

A remark related to Paul Levy's email on modules (Pat Donaly)

[Jpdonaly]

All categorists:

I can't respond to Paul Levy's request for sources, but the issue of
categorical modules may relate to a general question regarding the necessity of
V-enrichment via monoidal categories. So, in a backhanded sense, it could bear on
Paul's apparent search for something independent of standard V-enrichment.
Please pardon my naivete---my whole concern with this issue began just a few weeks
ago during some correspondence with Gabi Lukacs.

First, the connection, then the question: Since I am a little out of sympathy
with monoidal categories, if I want to enrich the homsets of a category R
into objects of some other category C which has a function-valued forgetful
functor U on it, I look for a bifunctor r:RxR'--->C (where R' is the opposite
category of R) for which the function composite functor U o r is the identity
adjunction on R. This adjunction is the bicomposition functor which sends a pair
(a,b) to the function z-->azb (which maps between the obvious homsets). If
s:SxS'--->C is another such C-enrichment or structure, then a C-structure morphism
from r to s is a functor from R to S which satisfies a certain property which
is stated in terms of the identity adjunctions which are at issue. It
frequently happens that C has a salient C-structure c:CxC'--->C of its own, in which
case I am inclined to call a C-structure morphism from r to c an r-module. Take
R to be the multiplicative monoid of a small ring, r to be simultaneous left
and right multiplication in R and C to be the category of small commutative
group homomorphisms to see that a ring is a C-structure and that an r-module is
the usual idea of an R-module in this case. I hope that this is what Paul's
question is about. I would like to see more information along these lines,
myself.

My question generally asks for the relationship between what is apparently
called V-enrichment and the idea just outlined. I can see that, if I fix an
R'-object in the right argument of a C-structure r, I get an r-module (a left
regular r-module, in fact), and, by taking left or co- adjoint functors of such
modules (if possible), I should get a tensor product concept which should define
a monoidal category composition for which the V-enrichment is r. The
literature has presumably examined the extent to which this is valid, and I would
appreciate being told where. Second, with my ingrained if idiosyncratic prejudices
against monoidal categories, I am curious to know if in some impressive sense
all (presumably closed?) monoidal categories come about in this way? Are
those which don't particularly interesting? Or is the situation the reverse: Every
worthwhile monoidal category comes from a C-structure r, but there are
important r's which don't provide a monoidal category. Is the full story laid out in
a book? Michael Kelley's book is out of print according to Gabi Lukacs. Any
help?

Pat Donaly