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/ ]