V-modules

V-modules

[Stefan Forcey]

Hello!
 First here are some preprints that relate delooping in topology to the
functor described by enriching over a monoidal category:

 http://arxiv.org/abs/math.CT/0304026  Enrichment as Categorical Delooping I: Enrichment Over Iterated Monoidal Categories

  http://arxiv.org/abs/math.CT/0306086 Higher Dimensional Enrichment

In continuing this research my collaborator and I have become interested
in extending results to V-modules (with hope of an explicit categorical
looping given by taking endofunctors becoming clear.) I notice in the
literature that there are two definitions of a V-module. Less common is a
definition that corresponds more closely to a classical module. This is
described as a category with a left (right) functorial action of V. Here
V-module functors and natural transformations are easily defined as well
(forming a 2-category Mod_V?) For instance, for c an object in C a
V-module, v in V, then a V-mod-functor F:C->D repects the action: F(vc) =
vF(c). V itself is a V-module in this sense, and End(V) = V seems to hold.
It also looks as though for V braided, left V-modules have a canonical
right structure, perhaps leading to a tensor product of V-V-bimodules. Any
references on this?
 The second definition is more common: for V closed, braided, a V-module
is a V-functor F:B^op tensor A -> V. These form the one-cells in a
bicategory V-Mod (objects V-categories, two-cells V-nat.trans.). Here
(given enough structure) we recover V as V-Mod(1,1) where 1 is the unit
V-category |1| ={0} and 1(0,0) = I the unit object of V. I noticed that
there may be a way to describe categories of these (Street's) V-modules as
(classical) V-modules. Modulo some careful checking, V-Mod(A,B) has an
action of V given by (vF)(A,B) = v tensor F(A,B). The details of this
action on morphisms require the adjunction that makes V closed.
 Is there some well-known deep connection between the two concepts that I
have missed due to my youth and naivete? We would be thankful for any
helpful comments or suggested references. Thanks,

 Stefan Forcey
VA Tech.