From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7216 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: when is Lex[A,V] abelian? Date: Sat, 25 Feb 2012 05:09:04 +0000 Message-ID: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1330170732 31669 80.91.229.3 (25 Feb 2012 11:52:12 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 25 Feb 2012 11:52:12 +0000 (UTC) To: Categories mailing list Original-X-From: majordomo@mlist.mta.ca Sat Feb 25 12:52:11 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1S1GAb-0001r6-6c for gsmc-categories@m.gmane.org; Sat, 25 Feb 2012 12:52:09 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:47940) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1S1G9M-0000X0-Bb; Sat, 25 Feb 2012 07:50:52 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1S1G9L-0004xV-Ni for categories-list@mlist.mta.ca; Sat, 25 Feb 2012 07:50:51 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7216 Archived-At: Hi Ignacio, If C has finite limits (in fact kernels would do) as well as finite colimits, then Lex[C^op, V] abelian actually implies C abelian. Indeed, the restricted Yoneda embedding C -> Lex[C^op, V] then preserves limits and finite colimits, and is fully faithful; whence C may be identified with a full subcategory of L closed under finite limits and finite colimits. But any such subcategory of an abelian category is itself abelian. This then means that Lex[C^op, V] abelian =3D> C abelian =3D> Lex[C^op, V] Grothendieck abelian. When V =3D Ab, this is sufficient to ensure that Lex[C^op, V] is lex-reflective in [C^op, V]. Off the top of my head, I don't know if the same is true when V =3D k-Mod; I feel like it might be necessary to assume that, for every f.p. flat k-module M, the functor M * (-) : C -> C preserves monomorphisms. Maybe that's automatic; I don't know enough algebra to say for sure. If C doesn't have kernels, then the situation is more interesting. I believe it should still be possible to give elementary conditions on C which are equivalent to L's being abelian. The point is that kernels do exist in L and so one can work out what it means for L to satisfy the kernel-cokernel exactness conditions for maps between representables in terms of structure in C. This will give some necessary conditions on C for L to be abelian. With any luck they will also be sufficient, though it might be necessary to consider a wider class of maps in L than merely those between representables. A relevant article, I think, is C. Centazzo, R.J. Wood An extension of the regular completion J. Pure Appl. Algebra, 175 (2002), pp. 93=96108 That deals with the non-additive context but the same ideas should apply. I am also wondering if Mike Shulman's work on unary sites is relevant; see his CT2011 slides. Richard On 22 February 2012 11:31, Ignacio Lopez Franco wrote: > Dear all, > may be some of the readers of this list will know the answer to the > following question. > > Let V be the category k-Mod for commutative ring k. > For a finitely cocomplete V-category C, when is L =3D Lex[C^{op},V] abeli= an? > > I know some cases: > 1. When C is abelian so is L. > 2. When C is a free completion under finite colimits of a small > =A0 =A0category, L is abelian (because it's equivalent to a presheaf > =A0 =A0V-category). > 3. L is reflective in the abelian [C^{op},V]. When the reflection is > =A0 =A0left exact L is abelian. However I don't any conditions that > =A0 =A0guaranty that the reflection is left exact. > > I would like to know some other conditions that ensure that L is > abelian, and perhaps an example where L is not abelian. > > Thanks > Ignacio > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]