From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5663 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: converse relations and distributors Date: Fri, 26 Mar 2010 09:20:10 +1100 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1269571795 3002 80.91.229.12 (26 Mar 2010 02:49:55 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 26 Mar 2010 02:49:55 +0000 (UTC) To: John Stell , categories Original-X-From: categories@mta.ca Fri Mar 26 03:49:51 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1NuzcM-00074z-1W for gsmc-categories@m.gmane.org; Fri, 26 Mar 2010 03:49:50 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Nuz68-0005Gq-5u for categories-list@mta.ca; Thu, 25 Mar 2010 23:16:32 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5663 Archived-At: Dear John, Let Dist be the category of posets and distributors (also known as profunctors, modules, ...) between them. Dist is monoidal via the categorical product of posets. This is not the product in Dist, but I will still write AxB for the product of A and B. Then the opposite poset of A (reverse the direction of inequalities) is dual to A, in the sense of monoidal categories: there are distributors 1--/--> A*xA and AxA*--/-->1 satisfying the triangle equations for an adjunction. It follows that any distributor A--/-->B induces a distributor B*--/-->A*, which one could call the converse (it is the converse as a relation). All this works for categories rather than posets, but then Dist has to be regarded as a monoidal bicategory rather than a monoidal category, and the notion of dual has to be suitably adapted. More generally it holds for enriched or internal categories, provided that the base for enrichment or internalization has enough structure. Regards, Steve Lack. On 26/03/10 12:38 AM, "John Stell" wrote: > The notion of a "distributor between posets" is used as a motivating example > in the document > Distributors at Work (www.mathematik.tu-darmstadt.de/~streicher/). This gives > one way to > see what a relation between posets should be. > > For posets A and B, what is considered are (ordinary) relations R \subseteq B > \times A such that > b2 \leq b1 and a1 \leq a2 and b1 R a1 implies b2 R a2. > > What I'm interested in is the converse of such a relation between posets. > Clearly we cannot simply > take the usual converse of R. I can see what to do for the context in which I > need this, but I'd > be interested to know if this issue appears in the literature. Is there some > well-known construction > on distributors that tells us what the converse 'should' be? > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]