From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7720 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: (In)accessible comonads and (non)Grothendieck toposes Date: Sat, 11 May 2013 16:26:52 +0100 (BST) Message-ID: References: Reply-To: "Prof. Peter Johnstone" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1368398892 12891 80.91.229.3 (12 May 2013 22:48:12 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 12 May 2013 22:48:12 +0000 (UTC) Cc: "categories@mta.ca" To: David Roberts Original-X-From: majordomo@mlist.mta.ca Mon May 13 00:48:11 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Ubf3q-00083b-Sh for gsmc-categories@m.gmane.org; Mon, 13 May 2013 00:48:11 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48800) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Ubf2d-00040t-Po; Sun, 12 May 2013 19:46:55 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ubf2e-000778-FN for categories-list@mlist.mta.ca; Sun, 12 May 2013 19:46:56 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7720 Archived-At: A particular case of this is in SGA4, IV 9.5.4, where it is shown that the category obtained by (Artin) glueing along a finite-limit- preserving functor between Grothendieck toposes is a Grothendieck topos iff the functor is accessible. It was Gavin Wraith, in JPAA 4 (1974), who first observed that Artin glueing is a particular case of forming a category of coalgebras (and therefore works for elementary toposes without the accessibility condition). Peter Johnstone On Thu, 9 May 2013, David Roberts wrote: > Hi all, > > I am just wondering where it was first stated (for both directions) that > the category of coalgebras for a comonad on a Grothendieck topos E is > again Grothendieck if and only if the underlying endofunctor of E is > accessible. > > A modern argument might go as: the topos of coalgebras is Grothendieck > if and only if it is locally presentable if and only if the endofunctor > is accessible, the original probably just mentioned preservation of > filtered colimits. > > Many thanks, > > David Roberts > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]