From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5711 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Four problems Date: Mon, 26 Apr 2010 11:09:36 -0400 Message-ID: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1272458507 18631 80.91.229.12 (28 Apr 2010 12:41:47 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 28 Apr 2010 12:41:47 +0000 (UTC) To: Original-X-From: categories@mta.ca Wed Apr 28 14:41:43 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 1O76aE-0001jB-3E for gsmc-categories@m.gmane.org; Wed, 28 Apr 2010 14:41:42 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1O761X-0006lG-Lm for categories-list@mta.ca; Wed, 28 Apr 2010 09:05:51 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5711 Archived-At: 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) >Prob2. A "little" bit harder, in the same vein. Let C be a (small) = category,=20 > S a set of maps of C and P: C --> C[Inv(S)] be the universal functor=20 > which inverts all maps of S. What conditions must the pair (C,S) = satisfy=20 >so that the functor >P is faithful? > If P: C --> S is a functor, I denote by V(P) the subcategory of C =20 >which has the same objects and as maps the vertical maps i.e. the f's = such=20 >that P(f) is an identity. Let V be a subcategory of a (small) category = C.=20 > What conditions >must the pair (C,V) satisfy in order that: >Prob3. There exists a functor P with domain C such that V =3D V(P) >Prob4. There exists a fibration P with domain C such that V =3D = V(P) Best, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]