From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8583 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Tensor product of left exact morphisms Date: Sat, 28 Mar 2015 19:40:12 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1427554478 4043 80.91.229.3 (28 Mar 2015 14:54:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 28 Mar 2015 14:54:38 +0000 (UTC) Cc: Categories list To: henry@phare.normalesup.org Original-X-From: majordomo@mlist.mta.ca Sat Mar 28 15:54:29 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 1Ybs7u-0006zs-Mf for gsmc-categories@m.gmane.org; Sat, 28 Mar 2015 15:54:18 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:58838) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Ybs6p-0001Xn-Gl; Sat, 28 Mar 2015 11:53:11 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ybs6o-0003lt-Ck for categories-list@mlist.mta.ca; Sat, 28 Mar 2015 11:53:10 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8583 Archived-At: Thanks, Simon. So it seems that, however one wishes to prove it (and in fact there are many ways), my original answer was correct. I subsequently tried to correct my original answer, but in fact it turns out that I corrected it to something else which was correct. Indeed, if: - X is the category of f.g. free G^op-sets (for G a group or a groupoid) - Y is the subcategory of X on the well-supported objects then [Y, Set], my first answer, is equivalent to Sh(X^op), my second answer, and both classify the notion of inhabited free right G-set. Here on the right we are taking sheaves for the topology on X^op comprising the project projections. The point is that: - every object of X is covered by one of Y; - if A is in Y, then the sieve in X generated by any product projection BxA-->A is easily seen to be the maximal one whence by the comparison lemma, sheaves on X^op are the same as presheaves on Y^op. Richard On Sat, Mar 28, 2015, at 12:08 AM, henry@phare.normalesup.org wrote: > 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 > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]