From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5722 Path: news.gmane.org!not-for-mail From: tholen@mathstat.yorku.ca Newsgroups: gmane.science.mathematics.categories Subject: Re: Four problems Date: Fri, 30 Apr 2010 21:13:59 -0400 Message-ID: References: Reply-To: tholen@mathstat.yorku.ca NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;format="flowed" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1272737977 27198 80.91.229.12 (1 May 2010 18:19:37 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 1 May 2010 18:19:37 +0000 (UTC) To: "Joyal, =?iso-8859-1?b?QW5kcuk=?=" , categories@mta.ca Original-X-From: categories@mta.ca Sat May 01 20:19:36 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 1O8HHr-0005EX-U4 for gsmc-categories@m.gmane.org; Sat, 01 May 2010 20:19:36 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1O8Gor-0007mB-DF for categories-list@mta.ca; Sat, 01 May 2010 14:49:37 -0300 In-Reply-To: Content-Disposition: inline Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5722 Archived-At: In the article J. Rosicky, W. Tholen, "Factorization, fibration and torsion", J. Homotopy Theory and Related Structures (electronic) 2 (2007) 295-314 we prove a result closely related to Problems 3 and 4 below (variations of which may well have appeared earlier?), as follows: In a finitely complete category C, (E,M) is a simple reflective factorization system of C (in the sense of Cassidy, Hebert, Kelly, J. Austr. Math. Soc 38, (1985)) if, and only if, there exists a prefibration P: C ---> B preserving the terminal object of C with E =3D P^{-1}(Iso B) and M =3D {P-cartesian morphisms}. Here "prefibration" means that for all objects c in C, the functors C/c ---> B/Pc induced by P have right adjoints, such that the induced monads are idempotent. (For a fibration one asks the counits to be identity morphisms.) Of course, Jean's question wants P^(-1)(Iso) to be replaced by the non-iso-closed class P^(-1)(Identities), which prevents the class from being part of an ordinary factorization system. But (without having looked into this at all) I would suspect that there is probably a (more cumbersome) reformulation of the theorem above which would address that concern. Regards, Walter. Quoting "Joyal, Andr=E9" : > Jean Benabou has formulated four problems of category theory. > They were communicated to a restricted list of peoples, not a private lis= t. > I see no serious raisons for not sharing these problems with everyones. > 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? >> (Generalized "Mal'cev" conditions) > >> Prob2. A "little" bit harder, in the same vein. Let C be a (small) >> category, >> S a set of maps of C and P: C --> C[Inv(S)] be the universal functor >> which inverts all maps of S. What conditions must the pair (C,S) satis= fy >> so that the functor >P is faithful? > >> If P: C --> S is a functor, I denote by V(P) the subcategory of C >> which has the same objects and as maps the vertical maps i.e. the f's su= ch >> that P(f) is an identity. Let V be a subcategory of a (small) category = C. >> 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/ ] > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]