From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2421 Path: news.gmane.org!not-for-mail From: Stefan Forcey Newsgroups: gmane.science.mathematics.categories Subject: Re: module for a category Date: Mon, 25 Aug 2003 13:55:41 -0400 (EDT) Message-ID: <20030825175543Z10615-24564+241@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 1241018648 4091 80.91.229.2 (29 Apr 2009 15:24:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:08 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Aug 26 14:30:17 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 26 Aug 2003 14:30:17 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19rhZk-0005cL-00 for categories-list@mta.ca; Tue, 26 Aug 2003 14:25:48 -0300 X-Mailer: ELM [version 2.5 PL2] Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 13 Original-Lines: 45 Xref: news.gmane.org gmane.science.mathematics.categories:2421 Archived-At: What you are looking for may be similar to something I queried Ross Street in regard to earlier this summer. I'll save him some time by putting here the relevant part of his response. > I think the one you first >mention is what we have been calling V-actegories. Benabou looked at >these rather than (as well as?) V-categories in the early days of >monoidal categories. Pareigis also made use of them. More recently, >publications of Paddy McCrudden involve them. There is a close >connection with V-categories. A V-module V x A --> A in this sense >for which we have a parametrized adjoint V(x,[a,b]) =~ A(x.a,b) >makes A a V-category with V-valued hom [a,b]. > >Conversely, a tensored V-category becomes such a V-module. I recommend the work of McCrudden, who has developed among other things a descent theoretic approach to the tensor product of V-actegories. There is also resource in the work of Harald Lindner. His paper, Enriched Categories and Enriched Modules, in Cahiers, Vol XXII-2 (1981) develops morphisms between enriched categories and actegories, which he calls modules. I'm curious about why it is that I have never seen his work referenced. Paul B Levy writes: > > Hi > > Is there a standard reference for the notion of "left module for a > category"? (or right module, or bimodule) > > Is there any reference in the setting of ordinary categories rather than > (or as well as) enriched categories or bicategories? > > Thanks > Paul > > > > >