Re: Tensor product of left exact morphisms

[Richard Garner <richard.garner@mq.edu.au>]

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


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