From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1986 Path: news.gmane.org!not-for-mail From: "Dr. P.T. Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Pro C Date: Thu, 31 May 2001 13:57:45 +0100 (BST) Message-ID: References: <200105300438.f4U4chS51110@transbay.net> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018258 1577 80.91.229.2 (29 Apr 2009 15:17:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:17:38 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu May 31 21:35:58 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f510InV30317 for categories-list; Thu, 31 May 2001 21:18:49 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f In-Reply-To: <200105300438.f4U4chS51110@transbay.net> X-Scanner: exiscan *155S1F-0001Yv-00*ZKzR26atu/k* http://duncanthrax.net/exiscan/ Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 61 Original-Lines: 22 Xref: news.gmane.org gmane.science.mathematics.categories:1986 Archived-At: On Tue, 29 May 2001, Bill Rowan wrote: > > I have read that if C is a category, and the axiom of choice is assumed, then > Pro C is equivalent to its full subcategory of diagrams where the diagram > category is an inversely-directed set. Does anyone know where this is proved > in the literature? > > Thanks, > > Bill Rowan > Choice isn't needed: all you need is the result that, for any filtered category C, there is a directed poset P and a final functor P --> C. There is a proof of this somewhere in SGA4 (I don't have the reference to hand), where it is attributed to Pierre Deligne; but I suspect it may be older than this. Peter Johnstone