From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8579 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: RE: Tensor product of left exact morphisms Date: Thu, 26 Mar 2015 15:00:05 +0000 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1427460093 8734 80.91.229.3 (27 Mar 2015 12:41:33 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 27 Mar 2015 12:41:33 +0000 (UTC) To: Richard Garner , Categories list Original-X-From: majordomo@mlist.mta.ca Fri Mar 27 13:41:27 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 1YbTZW-0003md-1I for gsmc-categories@m.gmane.org; Fri, 27 Mar 2015 13:41:10 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:57469) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1YbTYp-0005GR-U4; Fri, 27 Mar 2015 09:40:27 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1YbTYn-0002Z7-QE for categories-list@mlist.mta.ca; Fri, 27 Mar 2015 09:40:25 -0300 Thread-Topic: categories: Tensor product of left exact morphisms Thread-Index: AQHQZ7/6zYMn4iQjVk+TCWvs2FgJFp0u04uY In-Reply-To: Accept-Language: en-US, en-CA Content-Language: en-US Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8579 Archived-At: Dear Richard,=0A= =0A= You are almost right. Except that the notion of non-transitive torsor =0A= should be made explicit. I would say that a right G-set E=0A= is a *non-transitive torsor* if the action of G on E is free.=0A= Equivalently, if E is a G-torsor over E/G. =0A= With this notion, the classifying topos for right free G-action=0A= is the topos of *covariant* set valued functors on=0A= the category of finitely generated free G^op-sets.=0A= =0A= A non-transitive G-torsor E can be viewed as a family =0A= of G-torsors indexed by E/G. In general, if a topos $mathcal{E}$=0A= classifies the models of a geometric theory T, there is=0A= another topos $mathcal{E}$ which classifies variable families=0A= of models of T: it is the *bagdomain* of $mathcal{E}$=0A= introduced by Johnstone. See the Elephant vol. I Proposition 4.4.16.=0A= =0A= Best regards,=0A= Andr=E9=0A= =0A= =0A= =0A= =0A= ________________________________________=0A= From: Richard Garner [richard.garner@mq.edu.au]=0A= Sent: Wednesday, March 25, 2015 9:27 PM=0A= To: Categories list=0A= Subject: categories: Tensor product of left exact morphisms=0A= =0A= Dear categorists,=0A= =0A= If G is a group, then [G,Set] is the classifying topos for right=0A= G-torsors.=0A= =0A= What about the classifying topos for possibly non-transitive torsors?=0A= =0A= I'm not very adept at these calculations, but if I construct it as a=0A= subtopos of the classifying topos for G^op-sets, it appears to come out=0A= as [X,Set] where X is the category of finitely presentable, free,=0A= non-empty G^op-sets.=0A= =0A= Similarly, if G is a groupoid, the corresponding classifying topos=0A= appears to be [X,Set], where X is the full subcategory of [G^op, Set] on=0A= those finite coproducts of representables which have global support.=0A= =0A= Is this correct?=0A= =0A= Richard=0A= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]