From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7454 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Sheaves of T-algebras Date: Mon, 24 Sep 2012 07:38:29 -0400 (EDT) Message-ID: References: <50602B71.9010008@cs.bham.ac.uk> Reply-To: Michael Barr NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1348510452 24833 80.91.229.3 (24 Sep 2012 18:14:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Mon, 24 Sep 2012 18:14:12 +0000 (UTC) Cc: Categories list To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Mon Sep 24 20:14:17 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 1TGDAe-0006wz-5Q for gsmc-categories@m.gmane.org; Mon, 24 Sep 2012 20:14:16 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51943) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1TGD9y-0000op-An; Mon, 24 Sep 2012 15:13:34 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1TGD9x-0007JG-TS for categories-list@mlist.mta.ca; Mon, 24 Sep 2012 15:13:33 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7454 Archived-At: Well, I actually know quite a bit more about it. First off, each point in the base has a local neighborhood whose sections are algebras and the restrictions are homomorphisms. I should have mentioned that all ops and partial ops are finitary, but of increasing arity. We know the answer is positive; I am just unhappy with the proof. The situation is that the stalks are integral domains. Now integral domains are not models of an LE theory. But they are models of a theory with an infinitude of LE operations (all finitary, but of increasing arity) and I want the global sections to be models of that theory (which turns out to characterize the rings in the limit closure of the domains). The space is the set of primes of some arbitrary semi-prime (no nilpotents) commutative ring with the co-Zariski topology. Mike On Mon, 24 Sep 2012, Steve Vickers wrote: > 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 >> > -- The modern conservative is engaged in one of man's oldest exercises in moral philosophy--the search for a superior moral justification for selfishness. --J.K. Galbraith [For admin and other information see: http://www.mta.ca/~cat-dist/ ]