Subject: bimodules in a biclosed category

From: categories <[email-masked]>
Date: Thu, 18 Jan 1996 12:03:01 -0400 (AST)

Date: Thu, 18 Jan 1996 10:38:59 +1100
From: Ross Street <[email-masked]>

In response to Michael Barr's question:

Theorem: If  V  is a closed braided monoidal category which is complete
and cocomplete then the bicategory  V-Mod  of V-categories, V-modules
(sometimes called V-bimodules, V-distributors or V-profunctors), and
V-module morphisms is a monoidal bicategory (meaning the hom of a
tricategory with one object).  However, in order for  V-Mod  to be
braided,  V  must be symmetric in which case  V-Mod  is also
symmetric (also called strongly involutory by Baez-Dolan).

If you are only interested in monoids in  V,  just take the subbicategory
of one-object V-categories. Of course, then, you can do with less
completeness and cocompleteness on  V.

But yes, the detailed proof of this makes a long, but fairly routine,
story. Brian Day and I are working on a paper "Monoidal bicategories
and Hopf algebroids" which will contain a discreet amount of detail
(along with other things). I have been talking about aspects of the
paper in our Category Seminars.

Regards,
Ross