From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/284 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: RE: Finiteness in Toposes Date: Wed, 22 Jan 1997 14:41:21 -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 1241016869 25027 80.91.229.2 (29 Apr 2009 14:54:29 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:54:29 +0000 (UTC) To: categories Original-X-From: cat-dist Wed Jan 22 14:43:07 1997 Original-Received: by mailserv.mta.ca; id AA31635; Wed, 22 Jan 1997 14:41:21 -0400 Original-Lines: 43 Xref: news.gmane.org gmane.science.mathematics.categories:284 Archived-At: Date: Wed, 22 Jan 1997 18:18:53 +0000 From: Steve Vickers >Date: Fri, 17 Jan 1997 12:50:13 -0500 (EST) >From: F William Lawvere > >Re: Finiteness in Toposes Jan 17 1997 > >This concerns the possibility , mentioned in my previous message, of two >internal toposes of finite objects. > > The conjecture that there are two natural internal categories of >finite objects is partly supported by the fact that there are two natural >natural-numbers objects, the usual one N that parameterizes compositional >iteration and another semicontinuous one L with the following features: > >... > > This object L has been discussed for 25 years, but I dont know if >anyone published the working-out of its properties and role. Am I right in thinking this to be Idl N, the ideal completion of the natural numbers (with their usual order)? I conjecture that this is a suitable value domain for the ranks of matrices over localic fields such as the reals: rank^-1{n} is not open, but rank^-1{n, n+1, n+2, ...} is. Then rank A is the set of natural numbers n such that we can find enough apartnesses to prove linear independence of n rows of A, and this is an ideal of N - the definition also smoothly incorporates infinite matrices. (Perhaps this is just one of the things that have have been discussed for 25 years and I'm reiventing it.) Anyway, I have investigated Idl N as a fixpoint object (in the sense of Crole and Pitts) in the category of Grothendieck toposes (modulo 2-categorical niceties that I didn't investigate too closely) in a paper "Topical Categories of Domains". Steve Vickers.