From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4484 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Co-categories Date: Mon, 18 Aug 2008 10:58:47 +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 1241019975 13517 80.91.229.2 (29 Apr 2009 15:46:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:46:15 +0000 (UTC) To: Categories List Original-X-From: rrosebru@mta.ca Mon Aug 18 09:27:21 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 18 Aug 2008 09:27:21 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1KV3pH-0006L5-Fz for categories-list@mta.ca; Mon, 18 Aug 2008 09:27:11 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 19 Original-Lines: 40 Xref: news.gmane.org gmane.science.mathematics.categories:4484 Archived-At: I was expecting Peter Lumsdaine to reply to this, but perhaps he's away. In discussions with Steve Awodey and myself, Peter recently established the fact that every co-category in a pretopos is a co-equivalence relation; more specifically, the "co-domain" and "co-codomain" maps (sorry, but I can't see any other way to describe them) are the cokernel pair of a (unique) monomorphism (namely, their equalizer). Peter Johnstone On Tue, 12 Aug 2008, 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 > > >