From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7560 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Splitting epis by wishful thinking Date: Fri, 4 Jan 2013 15:04:51 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1357309252 6226 80.91.229.3 (4 Jan 2013 14:20:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 4 Jan 2013 14:20:52 +0000 (UTC) Cc: categories list To: Andrej Bauer Original-X-From: majordomo@mlist.mta.ca Fri Jan 04 15:21:05 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.32]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Tr88v-0004bk-8A for gsmc-categories@m.gmane.org; Fri, 04 Jan 2013 15:21:05 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:56836) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Tr816-0000OK-Qd; Fri, 04 Jan 2013 10:13:00 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Tr87D-00014Y-O3 for categories-list@mlist.mta.ca; Fri, 04 Jan 2013 10:19:19 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7560 Archived-At: Hi Andrej, This isn't exactly what you want, but it's along the right lines. Given a small category with cokernel pairs, one can construct another category with cokernel pairs in which all epis have been "freely" split. By this, I mean that a chosen section has been freely added to every epi in the category, even the ones that already had a section; thus the construction is not idempotent. Basically one uses the small object argument. Consider the category K of small categories with cokernel pairs, and functors preserving such. Let C be the free category with cokernel pairs containing an epi e: it can be obtained by first forming the free category with cokernel pairs on an arrow f, and then coinverting the codiagonal of f. Let D be the free category with cokernel pairs containing a section-retraction pair (i,p). There is an obvious map C --> D in K which sends e to p. Now some E in K satisfies the axiom of choice if and only if it is projective (has the weak right lifting property) with respect to this map C --> D. K is locally finitely presentable, and so the map C --> D generates via the small object argument a weak factorisation system (L,R) on it. By the above argument, the fibrant objects for (L,R) are those small categories with cokernel pairs satisfying the axiom of choice. If one uses the algebraic version of the small object argument, the fibrant replacement for this w.f.s. is a monad, S, say. The action of this monad on objects freely adjoins sections for all epis; its algebras are precisely the small categories with cokernel pairs with a chosen section for each epi. One can ask what happens if one drops the assumption of cokernel pairs. Consider the category Cat_ac, whose objects are small categories in which every epi comes equipped with a chosen section. There is an obvious forgetful functor Cat_ac ---> Cat, and a more precise formulation of your original question would be to ask if this functor has a left adjoint. This is unclear to me; at the moment I feel like it probably doesn't. What does seem clear is that, if it does have a left adjoint, then it can't possibly be monadic, so whatever construction one gives won't be entirely honest or straightforward. Richard On 3 January 2013 23:36, Andrej Bauer wrote: > On Mathoverflow there is a discussion (see > > http://mathoverflow.net/questions/117921/relative-consistency-of-etcs-over-the-theory-of-a-well-pointed-topos-with-nno > ) > which got me thinking. > > Is there a construction which "freely" splits all epis in a category > C? Something like: we add sections to every epi and wish we are done? > > The question is somewhat lose, but I think it is clear nontheless. > > With kind regards, > > Andrej > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]