From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2895 Path: news.gmane.org!not-for-mail From: Kirill Mackenzie Newsgroups: gmane.science.mathematics.categories Subject: Equivalence relations Date: Fri, 25 Nov 2005 13:41:37 +0000 (GMT) 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 1241018968 6258 80.91.229.2 (29 Apr 2009 15:29:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:28 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Nov 25 11:57:47 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 25 Nov 2005 11:57:47 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Effq0-0001kd-Vk for categories-list@mta.ca; Fri, 25 Nov 2005 11:50:13 -0400 X-X-Sender: pm1kchm@makar.shef.ac.uk Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 40 Original-Lines: 89 Xref: news.gmane.org gmane.science.mathematics.categories:2895 Archived-At: Here are a couple of pointers for double groupoids (strict, but with arbitrary side structures) : 1. Given a double groupoid S, say over groupoids H and V with common base M, the set of orbits for S over V is a groupoid over the set of orbits for H over M. This is set out in \3 of @ARTICLE{, author = {K.~C.~H. Mackenzie}, title = {Double {L}ie algebroids and second-order geometry, {I}}, journal = {Adv. Math.}, volume = 94, number = 2, pages = {180--239}, year = 1992, } It gives an elegant way of passing from an affinoid to the corresponding `butterfly diagram' or `Morita equivalence'. 2. General quotients for a groupoid G can be formulated in terms of congruences, which in turn are sub double groupoids of the double groupoid structure on G\times G. This (done with Philip Higgins) is in \S2.4 of @book {MR2157566, AUTHOR = {Mackenzie, Kirill C. H.}, TITLE = {General theory of {L}ie groupoids and {L}ie algebroids}, SERIES = {London Mathematical Society Lecture Note Series}, VOLUME = {213}, PUBLISHER = {Cambridge University Press}, ADDRESS = {Cambridge}, YEAR = {2005}, PAGES = {xxxviii+501}, ISBN = {978-0-521-49928-3; 0-521-49928-3}, MRCLASS = {58H05 (53D17)}, MRNUMBER = {MR2157566}, } Kirill ---------- Forwarded message ---------- Date: Wed, 23 Nov 2005 17:01:03 +1030 From: David Roberts To: Categories List Subject: categories: Equivalence relations Dear all, Considering the well known fact that an equivalence relation R on a set S gives a groupoid S_R with object set S, and the quotient of S by R is pi_0(S_R), has anyone done any work on "equivalence relations" on categories? Taking the skeleton of a cat is the prototypical example, but what I had in mind was a more "relative" construction. Given a groupoid enriched in categories, taking a sort of Pi_1 would give us a groupoid mod "equivalent morphisms". There is a smell of relative homotopy about, and I don't know enough in that area. I realise there are a couple of levels to this game, as evidenced by Kapranov and Voevodsky in their paper on 2-cats and the Zamolodchikov tetrahedron equations - do we take a "skeleton" at one or more dimensions? Any pointers appreciated ------------------------------------------------------------------------ -- David Roberts School of Mathematical Sciences University of Adelaide SA 5005 ------------------------------------------------------------------------ -- droberts@maths.adelaide.edu.au www.maths.adelaide.edu.au/~droberts www.trf.org.au