From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5052 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: flat topologies Date: Sat, 11 Jul 2009 16:41:17 +0100 (BST) Message-ID: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1247404888 27690 80.91.229.12 (12 Jul 2009 13:21:28 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 12 Jul 2009 13:21:28 +0000 (UTC) To: Szlachanyi Kornel , categories@mta.ca Original-X-From: categories@mta.ca Sun Jul 12 15:21:21 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1MPyzW-0003KD-3T for gsmc-categories@m.gmane.org; Sun, 12 Jul 2009 15:21:18 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1MPyBT-0002qT-UK for categories-list@mta.ca; Sun, 12 Jul 2009 09:29:35 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5052 Archived-At: Here's one way to look at it. A flat functor F: C --> Set corresponds to a point p of the presheaf topos [C^op,Set]. Given a Grothendieck topology J, p factors through the sheaf topos Sh(C,J) iff F carries J-covering sieves to epimorphic families. Thus the particular J you define is the largest for which p factors through Sh(C,J); equivalently, Sh(C,J) is the image (in the surjection--inclusion sense) of the geometric morphism p: Set --> [C^op,Set]. Hence a topos has a presentation of this kind iff it admits a surjective geometric morphism from Set; equivalently, iff it is (equivalent to) the category of coalgebras for a finite-limit-preserving accessible comonad on Set. Peter Johnstone ------------------- On Fri, 10 Jul 2009, Szlachanyi Kornel wrote: > Dear List, > > I have some questions on flat functors and Grothendieck topologies. > It is probably well-known that if F: C--> Set is a flat functor > on a small category then there is a Grothendieck topology on C > in which the covering sieves S on the object c consist of sets > of arrows to c such that {Fs|s in S} is jointly epimorphic on Fc. > > 1. Could you tell me a reference for this statement? > > 2. Is there a characterization of Grothendieck topologies that > arise in this way from a flat functor? (`flat topologies'?) > > 3. For what categories C will there be a flat functor inducing > the canonical topology on C? > > Thank you for any help. > > Kornel > > --------------------------------------------------- > Kornel Szlachanyi > Research Institute for Particle and Nuclear Physics > of the Hungarian Academy of Science > Budapest > --------------------------------------------------- > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]