From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2339 Path: news.gmane.org!not-for-mail From: Stefan Forcey Newsgroups: gmane.science.mathematics.categories Subject: V-modules Date: Fri, 6 Jun 2003 12:32:25 -0400 (EDT) Message-ID: <20030606163239Z10345-15495+90@calvin.math.vt.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018590 3666 80.91.229.2 (29 Apr 2009 15:23:10 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:23:10 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jun 6 17:22:34 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 06 Jun 2003 17:22:34 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ONgf-0004WN-00 for categories-list@mta.ca; Fri, 06 Jun 2003 17:19:45 -0300 X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 30 Original-Lines: 39 Xref: news.gmane.org gmane.science.mathematics.categories:2339 Archived-At: 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.