From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7014 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: when does preservation of monos imply left exactness? Date: Fri, 28 Oct 2011 23:36:40 +0200 Message-ID: References: Reply-To: "George Janelidze" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed;charset="UTF-8"; reply-type=original Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1319850725 23928 80.91.229.12 (29 Oct 2011 01:12:05 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 29 Oct 2011 01:12:05 +0000 (UTC) To: Original-X-From: majordomo@mlist.mta.ca Sat Oct 29 03:12:01 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 1RJxSp-0004sp-1h for gsmc-categories@m.gmane.org; Sat, 29 Oct 2011 03:11:59 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42968) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RJxRI-00053w-0M; Fri, 28 Oct 2011 22:10:24 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RJxRG-0001pi-0f for categories-list@mlist.mta.ca; Fri, 28 Oct 2011 22:10:22 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7014 Archived-At: Dear Dmitry, Let me write then F(X) = A@X, denoting the tensor product over k by @. Hence, briefly, @ is + (coproduct) in k-Alg. The fact that, when A is flat as a k-module, A+(-) : k-Alg --> k-Alg preserves finite limits is absolutely classical, as well as its proof via modules. However, you do not need the fully faithful functor from k-Mod to k-Alg given by the square-zero extension for that: simply use the fact the forgetful functor k-Alg --> k-Mod not only preserves limits, but also reflects isomorphisms (Some people prefer to say/use "creates isomorphisms"). The preservation of finite limits by A+(-) : k-Alg --> k-Alg (already for modules) implies, for example, Grothendieck's Descent Theorem saying that if A is flat, then k --> A is an effective descent morphism. However, much stronger result is known now: k --> A is an effective descent morphism if and only if it is pure as a monomorphism of k-modules. This stronger result, commonly known as an unpublished theorem of A. Joyal and M. Tierney, was claimed several times by various authors, and I can tell you more if you are interested. Anyway, since the category k-Alg has very bad exactness properties in a sense, modules were used by category-theorists themselves. And, in spite of nice results mentioned by Richard Garner and Steve Lack, I still suggest not to go through a generalization of the abelian case. Well, such a suggestion surely cannot be "universal", and if you need a better suggestion, you should tell us why are you doing this - if I may say so. Best regards, George Janelidze -------------------------------------------------- From: "Dmitry Roytenberg" Sent: Friday, October 28, 2011 2:27 PM To: "Steve Lack" Cc: "George Janelidze" ; ; "Categories list" Subject: categories: Re: when does preservation of monos imply left exactness? > Dear colleagues, > > First, let me thank everyone who has replied so far: I am learning a > lot from this discussion (not being a category theorist myself). Far > be it from me to make sweeping statements about general category > theory, I am merely interested in particular functors between > particular categories. It just seems that preservation of monos would > be easier to check than preservation of equalizers, which is why I am > looking for general criteria. > > Consider the following example. Let k-Alg denote the category of > commutative k-algebras over some ground ring k, and let > > F:k-Alg-->k-Alg > > denote taking coproduct with an object A. Then F preserves all > colimits and finite products; furthermore, if F preserves monos, it > preserves all finite limits (which is the case iff A is flat as a > k-module). The only proof I know takes advantage of the convenient > fact that the coproduct in k-Alg extends to the tensor product in the > abelian category k-Mod, as well as the existence of a fully faithful > functor from k-Mod to k-Alg given by the square-zero extension. If > anyone knows a proof which does not involve going to k-Mod, just using > some general properties of k-Alg, I'd be very happy to learn about it. > Notice that k-Alg is not co-Mal'cev, nor is every monomorphism > regular. > > Thanks again, > > Dmitry > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]