Re: Tensor product of left exact morphisms

[henry@phare.normalesup.org]

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/ ]