From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5620 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: Opposite of objects in a bicategory? Date: Mon, 08 Mar 2010 16:08:12 +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 1268079221 24143 80.91.229.12 (8 Mar 2010 20:13:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Mon, 8 Mar 2010 20:13:41 +0000 (UTC) To: David Leduc , categories Original-X-From: categories@mta.ca Mon Mar 08 21:13:35 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 1NojKZ-0004HJ-55 for gsmc-categories@m.gmane.org; Mon, 08 Mar 2010 21:13:35 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Noilc-0001R6-Jf for categories-list@mta.ca; Mon, 08 Mar 2010 15:37:28 -0400 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5620 Archived-At: Dear David, There is a way, but more structure, and possibly a change of perspective is required. I don't know how to describe opposites in terms of the category Cat. But if you work instead with the monoidal bicategory Prof/Mod/Dist (all three names are used) you can. I'll write A* for the opposite of a category A. I'll write Prof for the bicategory whose objects are (small) categories, and whose morphisms from A to B are functors A-->[B*,Set]. These are called profunctors (or modules or distributors) from A to B. They are composed using colimits; one easy way to see the composition is that up to equivalence, we can regard such profunctors as being cocontinuous functors [A*,Set]-->[B*,Set] and from this latter point of view we simply use ordinary composition of (cocontinuous) functors. Prof is monoidal, via the cartesian product of categories AxB. Note that this is not a cartesian monoidal structure on Prof, although it does have some features of cartesianness; it is an example of what is called a cartesian bicategory (studied by Carboni, Walters, Wood, and others). Anyway, in Prof, the opposite of a category B is dual to B, in the sense of monoidal (bi)categories, since functors AxB-->[C*,Set] correspond to functors A-->[BxC*,Set], and so to functors A-->[(CxB*)*,Set]. Thus in Prof, morphisms AxB-->C correspond to morphisms A-->CxB*. Steve Lack. On 7/03/10 11:28 PM, "David Leduc" wrote: > Dear all, > > The same way monads on categories can be generalized to monads on > objects of a bicategory, is there a way to generalize opposites of > categories to opposites of objects in a bicategory? > > David > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]