From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4597 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Non-cartesian categorical algebra Date: Wed, 17 Sep 2008 12:41:06 +1000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v753.1) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020048 13988 80.91.229.2 (29 Apr 2009 15:47:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:28 +0000 (UTC) To: Joost Vercruysse , Categories Original-X-From: rrosebru@mta.ca Thu Sep 18 10:34:25 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Sep 2008 10:34:25 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KgJYa-0005Zd-PL for categories-list@mta.ca; Thu, 18 Sep 2008 10:28:28 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 67 Original-Lines: 34 Xref: news.gmane.org gmane.science.mathematics.categories:4597 Archived-At: There is an embedding theorem on which we have put Cayley's name: if M is a monoid in a closed category then the structural coretraction M --> [M,M] into the endohom is a nice monoid map. A bicategorical version of this gives a nice module (distributor) A --|--> A^{op} #A for any (pro)monoidal V-category A. This leads to a monoidal embedding of any such A into the category of A-bimodules. (E.g. see Section 4 of Pastro-St: http://www.tac.mta.ca/tac/volumes/21/4/21-04.pdf however Brian Day also knew about these things.) So the abstract case is not so much more abstract. I think Peter Johnstone says somewhere that one view of the Abelian Category Embedding Theorem is not so much that it means we should use module-proofs to work in abelian categories but rather, when working in categories of modules, we might as well work in an abelian category. I think the same applies here for monoidal categories. The coring people I have spoken to seem quite comfortable with this development. Luckily we all have our own sources of motivation. Ross On 15/09/2008, at 10:57 PM, Joost Vercruysse wrote: > cocategory), corings provide examples of these internal cocategories, > but they (usually) refer to a much more concrete situation: a coring > is a co-monoid in the monoidal category of bimodules over a given > (possibly non-commutative) ring, this dualizes usual ring extensions.