From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4471 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Co-categories Date: Tue, 12 Aug 2008 20:45:38 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019967 13464 80.91.229.2 (29 Apr 2009 15:46:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:07 +0000 (UTC) To: Categories List Original-X-From: rrosebru@mta.ca Wed Aug 13 10:19:07 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 10:19:07 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTGEt-0001ji-NF for categories-list@mta.ca; Wed, 13 Aug 2008 10:18:11 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 6 Original-Lines: 25 Xref: news.gmane.org gmane.science.mathematics.categories:4471 Archived-At: I've been thinking idly about a concept dual to categories in much the same way that co-algebras are dual to algebras, and I've decided that I'd like to more about it. To be precise, if V is a monoidal category, then a category enriched over V has maps [A,B] (x) [B,C] -> [A,C], while a cocategory enriched over V has maps [A,C] -> [A,B] (x) [B,C]. (You can fill in the rest of the definition for yourself.) Searching Google, this concept appears to be known (under this name) in the case where V is Abelian, but I'm not so interested in that. I'm more interested in the case where V is a pretopos (like Set) equipped with the coproduct (disjoint union) as the monoidal structure (x). My motivation is that this concept is important in constructive analysis when V is a Heyting algebra equipped with disjunction as (x). (This defines a V-valued apartness relation on the set of objects; but I'm stating even this fact in more generality than I've ever seen.) So if anyone has heard of this concept where V is not assumed abelian, or even knows of a good introduction where V is assumed abelian, then I would be interested in references. --Toby