Re: Tensor product of left exact morphisms
Thanks, André, 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/ ]
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/ ]
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/ ]
participants (2)
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henry＠phare.normalesup.org -
Richard Garner