Dear Richard, You are almost right. Except that the notion of non-transitive torsor should be made explicit. I would say that a right G-set E is a *non-transitive torsor* if the action of G on E is free. Equivalently, if E is a G-torsor over E/G. With this notion, the classifying topos for right free G-action is the topos of *covariant* set valued functors on the category of finitely generated free G^op-sets. A non-transitive G-torsor E can be viewed as a family of G-torsors indexed by E/G. 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. Best regards, André ________________________________________ From: Richard Garner [richard.garner@mq.edu.au] Sent: Wednesday, March 25, 2015 9:27 PM To: Categories list Subject: categories: Tensor product of left exact morphisms Dear categorists, If G is a group, then [G,Set] is the classifying topos for right G-torsors. What about the classifying topos for possibly non-transitive torsors? I'm not very adept at these calculations, but if I construct it as a subtopos of the classifying topos for G^op-sets, it appears to come out as [X,Set] where X is the category of finitely presentable, free, non-empty G^op-sets. Similarly, if G is a groupoid, the corresponding classifying topos appears to be [X,Set], where X is the full subcategory of [G^op, Set] on those finite coproducts of representables which have global support. Is this correct? Richard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]