From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5718 Path: news.gmane.org!not-for-mail From: P.T.Johnstone@dpmms.cam.ac.uk Newsgroups: gmane.science.mathematics.categories Subject: Re: Four problems Date: 29 Apr 2010 11:16:54 +0100 Message-ID: References: Reply-To: P.T.Johnstone@dpmms.cam.ac.uk NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1272637008 8963 80.91.229.12 (30 Apr 2010 14:16:48 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 30 Apr 2010 14:16:48 +0000 (UTC) To: =?ISO-8859-1?Q?Andr=E9?= , categories@mta.ca Original-X-From: categories@mta.ca Fri Apr 30 16:16:47 2010 connect(): No such file or directory 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 1O7r1H-0005Ds-NR for gsmc-categories@m.gmane.org; Fri, 30 Apr 2010 16:16:43 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1O7qha-0006yv-8A for categories-list@mta.ca; Fri, 30 Apr 2010 10:56:22 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5718 Archived-At: On Apr 28 2010, Joyal, Andr=E9 wrote: >Jean Benabou has formulated four problems of category theory. >They were communicated to a restricted list of peoples, not a private list= . >I see no serious raisons for not sharing these problems with everyones.=20 >Here they are: > > >>Prob1. What conditions must a (small) category C satisfy in order that : >>there exists a faithful functor F: C --> G where G is a groupoid? = =20 >>(Generalized "Mal'cev" conditions) > This problem is solved in a recent paper of mine "On embedding categories in groupoids" in Math. Proc. Camb. Philos. Soc., vol. 145. Actually, the problem was essentially solved by Mal'cev and (independently) by Jim Lambek= , who gave necessary and sufficient conditions for a semigroup to be embeddable in a group; all one has to do is to observe that the same conditions work in the situation when one has several objects. Peter Johnstone [For admin and other information see: http://www.mta.ca/~cat-dist/ ]