Subject: More on Mike Barr's question

From: categories <[email-masked]>
Date: Wed, 24 Jan 1996 20:26:36 -0400 (AST)

Date: Thu, 25 Jan 1996 10:00:04 +1100
From: Ross Street <[email-masked]>

There has been some further discussion between Mike Barr and me
on modules between monoids in a complete, cocomplete, closed monoidal
category.

Mike responded:

>If I have understood your theorem, it is claimed only for the case that
>the original category is braided.  I was interested in the genuinely
>asymmetric case, so this doesn't apply.  Although your arguments
>are probably valid in that case, at a guess. Have I missed something?

So I said:

It looks as though **I** missed something. I didn't include the older
fact well known to enriched category theorists: even without any symmetry
or braiding,  V-Mod is a bicategory in which all right extensions and
right liftings exist (I don't like the word "biclosed"; I use "left closed",
"right closed" and "closed" for both when dealing with a monoidal category).
In particular, for any V-category  A, each hom category  V-Mod(A,A)  is a
closed monoidal category under composition (= tensor product) of modules
(as before, I don't say "bimodules" either). Even more particularly you
can take A to be a monoid in V.

For this there ARE references. I believe Benabou, in his Louvain-la-neuve
notes

"Les Distributeurs" Rapport No 33 jan 1973,

only looked at the case of V symmetric. Same is true of Bill Lawvere's
"Metric spaces" paper. But it is not hard to generalise. My paper

Enriched categories and cohomology, Quaestiones Math. 6 (1983)
265-283; MR85e:18007

does the non-symmetric case. But it does it more generally for  V
a bicategory (without commutativity you might as well have "several objects").
Other papers which build on this are:

Cauchy characterization of enriched categories, Rendiconti del Seminario
Matematico e Fisico di Milano  51 (1981) 217-233; MR85e:18006.

(with R. Betti, A. Carboni, and R. Walters) Variation through enrichment,
J. Pure Appl. Algebra 29 (1983) 109-127; MR85e:18005.

(with A. Carboni, S. Johnson and D. Verity) Modulated bicategories, J. Pure
Appl. Algebra 94 (1994) 229-282.

Gordon and Power have also done Gabriel-Ulmer duality for W-Mod where
W is a decent bicategory (homs locally presentable).  JPAA?

The recent thing with Brian Day looks at when  V-Mod itself is "monoidal",
a natural step beyond the above references & studied by Carboni, Walters
et al in the case where V is a poset (or W is locally a poset).

Best wishes,
Ross