From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/1991 Path: news.gmane.org!not-for-mail From: "Prof. T.Porter" Newsgroups: gmane.science.mathematics.categories Subject: Re: Pro C Date: Fri, 01 Jun 2001 09:48:50 +0100 Message-ID: <3B1756F2.82816D98@bangor.ac.uk> References: <200105300438.f4U4chS51110@transbay.net> <20010531074243.A26393@triples.math.mcgill.ca> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018261 1605 80.91.229.2 (29 Apr 2009 15:17:41 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:17:41 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Jun 1 06:47:19 2001 -0300 Return-Path: Original-Received: (from Majordom@localhost) by mailserv.mta.ca (8.11.1/8.11.1) id f5197Wp14872 for categories-list; Fri, 1 Jun 2001 06:07:32 -0300 (ADT) X-Authentication-Warning: mailserv.mta.ca: Majordom set sender to cat-dist@mta.ca using -f X-Mailer: Mozilla 4.7 [en] (X11; I; FreeBSD 3.3-RELEASE i386) X-Accept-Language: en, fr Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 1 Original-Lines: 36 Xref: news.gmane.org gmane.science.mathematics.categories:1991 Archived-At: William Boshuck wrote: > > This is due to Deligne, and can be found towards > the beginning of SGA4, Expose I, section 8. I would > like to know of a more recent source that is so (or > more) thorough on the subject. > cheers, > -b > On Tue, May 29, 2001 at 09:38:43PM -0700, 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 Dear All I replied to Bill Rowan directly yesterday but it now seems that others might be interested in my reply so here it is. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> In my book with Cordier, the result you want is Proposition 4 p 42 (The book is :Categorical Shape Theory, Cordier and Porter, Published by Ellis Horwood, 1989). The result is known to some shape theorists as the Mardesic trick as Sibe Mardesic is thought to have found it, but I seem to remember seeing a version of it in Grothendieck's work (SGA4 and earlier) If you can get a copy of our book there is a reasonably categorical treatment of pro categories. Best wishes, Tim Porter