From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/518 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Abelian-topos (AT) categories Date: Wed, 5 Nov 1997 17:35:26 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017012 26095 80.91.229.2 (29 Apr 2009 14:56:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:56:52 +0000 (UTC) To: categories Original-X-From: cat-dist Wed Nov 5 17:35:31 1997 Original-Received: by mailserv.mta.ca; id AA15826; Wed, 5 Nov 1997 17:35:26 -0400 Original-Lines: 27 Xref: news.gmane.org gmane.science.mathematics.categories:518 Archived-At: Date: Wed, 5 Nov 1997 11:11:52 +0000 From: Steven Vickers Under one view, there is a mismatch in the comparison between toposes and Abelian categories. Consider enriched category theory over Set and Ab[elian groups]. Over Set: enriched category A = small category, A-action = functor from A to Set (covariant or contra- for right or left action), cat of A-actions = Set^A or Set^A^op, wlog a presheaf topos, "quotient" (by Grothendieck topology) = general Grothendieck topos. Over Ab: enriched category A = ringoid ("ring with several objects"), A-action = right or left module over A, cat of A-actions = Mod-A or A-Mod, "quotient" (by Gabriel topology, a.k.a. hereditary torsion theory) = Grothendieck category, i.e. cocomplete Abelian category in which direct limits are exact and there is a generator. By the Lubkin-Heron-Freyd-Mitchell theorems, Abelian categories embed fully faithfully in Grothendieck categories but are more general. Assuming this parallel Grothendieck toposes || Grothendieck categories is a good one, is there a natural parallel of Abelian categories on the Set-enriched side? Steve Vickers.