From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7564 Path: news.gmane.org!not-for-mail From: Ohad Kammar Newsgroups: gmane.science.mathematics.categories Subject: Re: Splitting epis by wishful thinking Date: Fri, 4 Jan 2013 15:22:29 +0000 Message-ID: References: Reply-To: Ohad Kammar NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: ger.gmane.org 1357344600 2045 80.91.229.3 (5 Jan 2013 00:10:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 5 Jan 2013 00:10:00 +0000 (UTC) To: Richard Garner , Andrej Bauer , Categories List Original-X-From: majordomo@mlist.mta.ca Sat Jan 05 01:10:15 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 1TrHL0-0002NE-U4 for gsmc-categories@m.gmane.org; Sat, 05 Jan 2013 01:10:11 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58026) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1TrHEY-0001p7-D9; Fri, 04 Jan 2013 20:03:30 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1TrHKh-0003PM-25 for categories-list@mlist.mta.ca; Fri, 04 Jan 2013 20:09:51 -0400 In-Reply-To: Content-Disposition: inline Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7564 Archived-At: Dear Andrej, The following formulation of your question has negative answers: Let Cat be the category of small categories and functors between them, and SCat be its full sub-category of small categories in which all epis split. Let U : SCat -> Cat be the full inclusion. U does not have a left adjoint, nor a right adjoint. Proof: Let C be the following category: It has 3 objects: 0 - an initial object A and B. Apart from the identities and the initial maps, C has the following 3 morphisms: a parallel pair f, g : A -> B an endomorphism h : B -> B The non-trivial compositions are given by: x o h = x, for x = f, g, and h. Note that C is in SCat, as it has no trivial epimorphisms: the non-trivial arrow into A equalises f and g, which are different, and the non-trivial arrows into B equalise h and id. Let F : C -> C be the endofunctor that swaps f with g. If U had a left adjoint, then SCat was a full reflective subcategory of Cat, it was complete. Consider the equaliser of F, Id : C->C. Whatever it is in SCat, this coequaliser cannot be preserved by U, as the equaliser in Cat is given by dropping f and g. This resulting subcategory has an epi, the initial arrow into A, that doesn't split. Similarly, if U had a right adjoint, then SCat was cocomplete. Then the coequaliser of F, Id : C->C in Cat would be C with g dropped (=identified with f). But then the initial map into A becomes epi, without a section. Thus this colimit is not in SCat, and U doesn't preserve it. The same construction actually lies within the category Cat_ac that Richard described (with specified sections), as C has no non-trivial sections. Thus the same proof applies to his formulation, and his forgetful functor also isn't a right nor a left adjoint. Ohad. On 4 January 2013 04:04, Richard Garner wrote: > 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 >> > -- The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]