From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2442 Path: news.gmane.org!not-for-mail From: Krzysztof Worytkiewicz Newsgroups: gmane.science.mathematics.categories Subject: Re: associated sheaf functor Date: Wed, 17 Sep 2003 13:41:25 +0200 Message-ID: <3F684865.4040808@bluewin.ch> References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018665 4184 80.91.229.2 (29 Apr 2009 15:24:25 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:24:25 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Sep 17 12:18:07 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Sep 2003 12:18:07 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19ze1n-00073p-00 for categories-list@mta.ca; Wed, 17 Sep 2003 12:15:35 -0300 X-Accept-Language: en-us, en Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 10 Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:2442 Archived-At: > I don't quite understand this question. > I was interested since I am looking at the plus construction as part > of my work at the moment. Let P be a presheaf on the site (C,J) and consider the "classical" plus construction P^+(c) = colim_{R \in J(c)}Match(R,P) where Match(R,P) is the set of matching families for the cover R \in J(c) and the colimit is taken over J(c) ordered by reverse inclusion (cf. McLane & Moerdijk) . This is a nice filtered colimit so x \in P^+(c) can be expressed as an equivalence class of matching families. Suppose now that J is given by a basis K. It is not immediately clear (at least not for me) what happens in a variant of the above where Match(R,P) is taken as the set of matching families for the K-cover R. Indeed, the notion of "common refinement" for K-covers is not as handy as the one for J-covers for the task at hand since op-ordering K-covers will not necessarily give a filtered category. The other obvious candidate for a "category K(c)" where a factorisation witnessing a refinement (of K-covers) is a morphism (in the opposite category) will probably fail to be filtered as well, so I wondered if anybody allready looked at such things. I agree that a one-sentence prose might have been a bit messy... Cheers Krzysztof