From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4581 Path: news.gmane.org!not-for-mail From: Joost Vercruysse Newsgroups: gmane.science.mathematics.categories Subject: Re: Non-cartesian categorical algebra Date: Mon, 15 Sep 2008 14:57:55 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v926) Content-Type: text/plain; charset=US-ASCII; format=flowed; delsp=yes Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020039 13928 80.91.229.2 (29 Apr 2009 15:47:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:19 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Mon Sep 15 19:22:04 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Sep 2008 19:22:04 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KfMKr-0003Wj-9Y for categories-list@mta.ca; Mon, 15 Sep 2008 19:14:21 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 51 Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:4581 Archived-At: On 14-sep-08, at 15:39, George Janelidze wrote: Dear George and all, > There are also things-to-be-corrected happening: > for instance by far not enough comparisons have been made with the > Australian work on abstract monoidal categories, and some authors > use words > like "coring"... I hope the following information can be of help here: Indeed, Marcello Aguilar gave a definition of `internal categories'. Although the abstract definition of a `coring' looks formally the same as the one of an internal category (or, if you wish, an internal 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. The theory of corings is in fact quite young, and grew from a pure algebraic theory to something more and more categorical in the last few years (this might cause some confusion, `internal corings', which can be defined in certain monoidal categories (the regular ones from aguilar) or bicategories, are indeed the same objects as internal cocategories, there is no need for two names for the same thing at this level of generality). Therefore, I find the above remark ``not enough comparision have been made ...'' indeed correct: I believe that people from corings can learn from more from the pure category theory side, and hopefully the other way around as well. Best wishes, Joost.