From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8581 Path: news.gmane.org!not-for-mail From: henry@phare.normalesup.org Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor product of left exact morphisms Date: Fri, 27 Mar 2015 14:08:18 +0100 Message-ID: References: Reply-To: henry@phare.normalesup.org NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: 8bit X-Trace: ger.gmane.org 1427497981 15648 80.91.229.3 (27 Mar 2015 23:13:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 27 Mar 2015 23:13:01 +0000 (UTC) Cc: "Categories list" To: "Richard Garner" Original-X-From: majordomo@mlist.mta.ca Sat Mar 28 00:12:48 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1YbdQj-0003y6-2e for gsmc-categories@m.gmane.org; Sat, 28 Mar 2015 00:12:45 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58740) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YbdPu-0007Jb-OW; Fri, 27 Mar 2015 20:11:54 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YbdPt-0000g0-PH for categories-list@mlist.mta.ca; Fri, 27 Mar 2015 20:11:53 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8581 Archived-At: Dear Richard, If I'm not mistaken, the distinction between inhabited and non-inhabited torsors does not change much : your initial answer is correct. To connect with Andr?'s answer, inhabited free G-sets are inhabited collection of G-torsors, his construction produces a topos over [set,Set] (the classyfing topos for object, i.e. the bagdomain construction for the topos of sets) while what you want is a topos over [set+,Set] where set denote the category of finite set and set+ the category of inhabited finite set. But this corresponds to the functor from finitely generated free right $G$-Set to set which send an object to its (finite) set of orbits. Because it is a fibration it is easy to construct the pullback along the geometric morphism from [set+,Set] to [set,Set] corresponding to the inclusion of set+ in set : it will give the topos of inhabited free finitely generated G-set as you first found (the weak pullback of category). Also, as you are probably aware, once you know that the classyfing topos you want to construct is a topos of presheaf over a category C, it is a general fact that C can be taken to be the opposite of the category of finitely presented model of your theory, hence finitely generated free inhabited G-set, and what you said for the case of groupoids. Best wishes, Simon Henry > Thanks, Andr?, that's helpful. This: > >> In general, if a topos $mathcal{E}$ >> classifies the models of a geometric theory T, there is >> another topos $mathcal{E}$ which classifies variable families >> of models of T: it is the *bagdomain* of $mathcal{E}$ >> introduced by Johnstone. See the Elephant vol. I Proposition 4.4.16. > > is particularly good. I knew about the bagdomain, but didn't connect it > to my question. > > However, I think I want a non-transitive torsor to be a right G-set with > a free action, but which is also inhabited. This means passing to a > subtopos of [X, Set], where X is as before the category of finitely > generated free G^op sets. > > Looking at the calculation I made before, I think I got it wrong. I must > pass to the topology generated by making 0 ----> G into a cocover in X. > But then I must also make every pushout of this into a cocover, and > every composite of such pushouts into a cocover. So, in the end, I think > the classifying topos should be Sh(X^op) for the topology whose cocovers > are the coproduct injections in X. In other words, I take the Lawvere > theory of G^op-sets, and take sheaves on it for the topology given by > the project projections. > > Richard > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]