From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7012 Path: news.gmane.org!not-for-mail From: Steve Lack Newsgroups: gmane.science.mathematics.categories Subject: Re: when does preservation of monos imply left exactness? Date: Fri, 28 Oct 2011 09:08:14 +1100 Message-ID: References: Reply-To: Steve Lack NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1319806361 11210 80.91.229.12 (28 Oct 2011 12:52:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 28 Oct 2011 12:52:41 +0000 (UTC) Cc: "Dmitry Roytenberg" , "Categories list" To: "George Janelidze" Original-X-From: majordomo@mlist.mta.ca Fri Oct 28 14:52:36 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RJlvH-0006ia-Ib for gsmc-categories@m.gmane.org; Fri, 28 Oct 2011 14:52:35 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:53410) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RJluE-0001Zd-NS; Fri, 28 Oct 2011 09:51:30 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RJluD-0007G4-15 for categories-list@mlist.mta.ca; Fri, 28 Oct 2011 09:51:29 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7012 Archived-At: Dear All, While I agree that category theory is not just generalized abelian=20 category theory, there are nonetheless some theorems along the=20 lines that Dmitry suggests. Richard mentioned one; here is another. Of course neither is nearly as simple as the abelian case.=20 If the opposite of the categories A and B are exact Mal'cev, then any functor f:A->B which preserves finite colimits and regular monos = preserves equalizers of coreflexive pairs. Thus any functor f:A->B which preserves finite = colimits, finite products and regular monos preserves all finite limits. In particular, A and B could be toposes, then if f:A->B preserves finite = colimits, finite products, and monomorphisms, it preserves all finite limits.=20 =46rom the point of view of George's example, the problem is that in the = category 2, the map 0->1 is mono but not regular mono.=20 Best wishes, Steve Lack. On 27/10/2011, at 9:32 PM, George Janelidze wrote: > All right, somebody should answer... >=20 > Dear Dmitry, >=20 > First, may I suggest that looking at category theory as merely a > 'generalized abelian category theory' is not really a good idea? >=20 > Specifically, in the abelian context >=20 > 'right exact + preserves monos =3D> left exact' >=20 > is simple but extremely important, while for general categories it is = indeed > wrong, and adding conditions to make this work seems to be a strange = thing > to do! >=20 > Now, more specifically, concerning Barr exact: >=20 > Let 2 be the ordered set {0,1} considered as a category, let S be the > category of sets, and let F : S --> 2 be the left adjoint of the = canonical > embedding 2 --> S (hence, for a set X, F(X) =3D 0 if and only if X is = empty). > Then: >=20 > (a) 2 and S are Barr exact. >=20 > (b) F preserves all colimits since it is a left adjoint. >=20 > (c) F preserves products (moreover, the fact that it preserves all = products > is one of the standard forms of the Axiom of Choice). >=20 > (d) F preserves monomorphisms since every morphism in 2 is a = monomorphism. >=20 > (e) F does not preserve all equalizers since two parallel morphisms = between > non-empty sets might have the empty equalizer. >=20 > Best regards >=20 > George Janelidze >=20 > -------------------------------------------------- > From: "Dmitry Roytenberg" > Sent: Wednesday, October 26, 2011 1:59 PM > To: "Categories list" > Subject: categories: Re: when does preservation of monos imply left > exactness? >=20 >> I'll try again... >>=20 >> On Mon, Oct 24, 2011 at 11:23 AM, Dmitry Roytenberg >> wrote: >>> Dear category theorists, >>>=20 >>> It is well known that any functor that preserves finite limits >>> preserves monomorphisms, and that for an additive right-exact = functor >>> between abelian categories, the converse is also true. Is it known = how >>> far this extends to the non-additive setting? In other words, what >>> exactness properties of two categories and a functor between them >>> would suffice to conclude that the functor preserves finite limits = if >>> and only if it preserves monos? For instance, is it enough to assume >>> that the categories be Barr-exact and that the functor preserve all >>> colimits, finite products and monos to conclude that it also = preserves >>> equalizers? >>>=20 >>> Any references would be extremely helpful. >>>=20 >>> Thanks, >>> Dmitry >>>=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]