RE: Tensor product of left exact morphisms

["Joyal, André" <joyal.andre@uqam.ca>]

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


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