From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7453 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Sheaves of T-algebras Date: Mon, 24 Sep 2012 10:44:17 +0100 Message-ID: References: Reply-To: Steve Vickers NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1348486456 30000 80.91.229.3 (24 Sep 2012 11:34:16 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 24 Sep 2012 11:34:16 +0000 (UTC) Cc: Categories list To: Michael Barr Original-X-From: majordomo@mlist.mta.ca Mon Sep 24 13:34:21 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.134]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1TG6vc-0006ch-Tr for gsmc-categories@m.gmane.org; Mon, 24 Sep 2012 13:34:21 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51717) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1TG6vH-0007PM-NG; Mon, 24 Sep 2012 08:33:59 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1TG6vH-0006AN-CI for categories-list@mlist.mta.ca; Mon, 24 Sep 2012 08:33:59 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7453 Archived-At: Dear Mike, Once you have that a sheaf of T-algebras is equivalent to a T-algebra in the category of sheaves, it's obvious: because the global sections functor preserves finite (in fact all) limits. But the first step is not so trivial, and in fact is not true unless you take care over "the stalks are T-algebras". For example, if the base space is Sierpinksi then a sheaf is just a function, with the domain and codomain as the stalks (over the bottom and top points respectively). You might put algebra structures on the stalks, but it won't be an algebra in the sheaf category unless the function is a homomorphism. You will get further issues if the base space is non-spatial, so there aren't enough global points to provide enough stalks. What statement of the result did you have in mind? One sufficient condition that gets round those problems is for the stalks and their T-algebra structure to be defined geometrically. Regards, Steve. Michael Barr wrote: > It must be known that if T is a left exact theory (same as nearly > equational theory) and you make a sheaf of T-algebras (meaning the stalks > are T-algebras, then the set of global sections is also a T-algebra. Can > anyone give me reference? > > Michael > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]