From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4330 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: The internal logic of a topos Date: Mon, 17 Mar 2008 14:40:32 +0000 (GMT) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019875 12790 80.91.229.2 (29 Apr 2009 15:44:35 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:44:35 +0000 (UTC) To: Categories mailing list Original-X-From: rrosebru@mta.ca Mon Mar 17 18:15:34 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 17 Mar 2008 18:15:34 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JbMc6-0005XX-97 for categories-list@mta.ca; Mon, 17 Mar 2008 18:11:22 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 94 Original-Lines: 41 Xref: news.gmane.org gmane.science.mathematics.categories:4330 Archived-At: Dear Vaughan, I don't think one can give a straight answer to this question: it all depends on what you mean by `the logic of a topos'. I presume you're thinking of the fact that, in Set, any function 2^n --> 2 is a polynomial in the Boolean operations (i.e., is the interpretation of some n-ary term in the theory of Boolean algebras). One could ask the same question about a general topos, with `Heyting' replacing `Boolean'; but the answer would mostly be `no', even for Boolean toposes. On the other hand, one might well *define* `the internal logic of a topos' as meaning the collection of all natural operations on subobjects -- that is, the collection of all morphisms \Omega^n --> \Omega. Incidentally, there is nothing unnatural or counterintuitive about `the notion of internal Heyting algebra: it is a very natural consequence of the definition of a subobject classifier, see A1.6.3 in the Elephant. Peter On Sun, 16 Mar 2008, Vaughan Pratt wrote: > As I understand the internal logic of a topos it consists of certain > morphisms from finite powers of Omega to Omega. In the case of Set it > consists of all such morphisms. For which toposes is this not the case, > and for those how are the morphisms that do belong to the internal logic > best characterized? > > I do hope it's not necessary to start from the notion of an internal > Heyting algebra, that sounds so counter to mathematical practice and > intuition. > > If the internal logic consists of precisely those morphisms preserved by > geometric morphisms this will give me the necessary motivation to go to > the mats with geometry. > > Vaughan > > >