From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4475 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Co-categories Date: Wed, 13 Aug 2008 17:29:23 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019969 13478 80.91.229.2 (29 Apr 2009 15:46:09 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:09 +0000 (UTC) To: Categories List Original-X-From: rrosebru@mta.ca Wed Aug 13 20:02:52 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 13 Aug 2008 20:02:52 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KTPMF-0000JH-4S for categories-list@mta.ca; Wed, 13 Aug 2008 20:02:23 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 10 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:4475 Archived-At: Dear Toby, What you call a cocategory enriched over V can also be described as a category enriched over V^op. These have been studied by Paddy McCrudden in his thesis under the name "coalgebroids" (= many-object coalgebras). The main results are on a generalised notion of Tannakian duality; and on transfer of extra structure across this duality. See respectively: [1] Paddy McCrudden, Categories of Representations of Coalgebroids, Advances in Mathematics Volume 154, Issue 2, Pages 299-332 [2] Paddy McCrudden, Balanced Coalgebroids Theory and Applications of Categories, Vol. 7, pp 71-147. Richard --On 12 August 2008 20:45 Toby Bartels wrote: > 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 > > >