From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7457 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Sheaves of T-algebras Date: Tue, 25 Sep 2012 10:42:28 +0100 (BST) Message-ID: References: <50602B71.9010008@cs.bham.ac.uk> Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1348619268 6067 80.91.229.3 (26 Sep 2012 00:27:48 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 26 Sep 2012 00:27:48 +0000 (UTC) Cc: Steve Vickers , Categories list To: Michael Barr Original-X-From: majordomo@mlist.mta.ca Wed Sep 26 02:27:53 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 1TGfTj-0007dB-Aq for gsmc-categories@m.gmane.org; Wed, 26 Sep 2012 02:27:51 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54502) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1TGfTO-0006i4-Cp; Tue, 25 Sep 2012 21:27:30 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1TGfTP-00057F-Ao for categories-list@mlist.mta.ca; Tue, 25 Sep 2012 21:27:31 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7457 Archived-At: Dear Mike, In that case you want to look at John Kennison's 1976 paper "Integral domain type representations in sheaves and other topoi", Math. Z. 151, 35-56, and my own "Rings, fields and spectra" in J. Algebra 49 (1977), 238-260. There's a summary in section V 3.11 of "Stone Spaces". Peter Johnstone On Mon, 24 Sep 2012, Michael Barr wrote: > 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 >>> >> > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]