From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2860 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Two topos questions Date: Wed, 2 Nov 2005 21:22:46 +0000 (GMT) Message-ID: References: <2cc0d36c0511012253p1a15630ay1c0eb35f724905db@mail.gmail.com> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018948 6121 80.91.229.2 (29 Apr 2009 15:29:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:29:08 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Nov 2 20:29:27 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 02 Nov 2005 20:29:27 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EXSqG-0005yD-Ve for categories-list@mta.ca; Wed, 02 Nov 2005 20:20:33 -0400 In-Reply-To: <2cc0d36c0511012253p1a15630ay1c0eb35f724905db@mail.gmail.com> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 5 Original-Lines: 44 Xref: news.gmane.org gmane.science.mathematics.categories:2860 Archived-At: On Wed, 2 Nov 2005, Peter Arndt wrote: > Hi, category theorists, > 1. In a message to the categories list from 15. jan.1997 (that message can > be seen at http://www.mta.ca/~cat-dist/catlist/1999/finite-topos) Lawvere > talks about "the ... internal topos ... which parametrizes the decidable > K-finites". Does anyone know what exactly is that internal topos? Is there > some morphism that can be seen as the indexed family of decidable K-finites > (just like the generic cardinal "is" the indexed family of finite cardinals > and can be used to construct the full internal subcategory of finite > cardinals)? I can't remember exactly what Bill was talking about in that posting. However, there is no hope of `parametrizing' decidable K-finite objects by an internal category, unless the ambient topos has a natural number object (cf. the remarks on pp. 1058-9 of "Sketches of an Elephant"), and if it does the decidable K-finites are exactly the objects locally isomorphic to finite cardinals. So I suspect that he was referring to the internal category of finite cardinals. > 2. An object Y of a topos is said to have locally a property P if there is > an object Z with global support such that Z*(Y) has the property P. For the > topos of sheaves on a T1-space X (and a property P stable under pullback > along subterminals), I convinced myself that this implies the existence of a > covering of X, such that P holds on the restriction of Y to each open set of > the covering. Can this also be proved for schemes or other classes of > topological spaces, maybe with additional conditions on P? Yes, of course -- this is exactly the geometric intuition behind this use of "locally". One needs to assume that P is stable under arbitrary pullback (which will certainly be the case if it's expressible in the internal language of a topos). Then, in any topos generated by subterminals (in particular, in any topos of sheaves on a space), every cover Z -->> 1 is dominated by one of the form \coprod_i U_i -->> 1, where the U_i are a family of subterminals covering 1 in the classical sense. So P holds locally for Y iff it holds for the restriction of Y to each member of some cover in the classical sense. Peter Johnstone