Tensor product of left exact morphisms

Tensor product of left exact morphisms

[Richard Garner]

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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RE: Tensor product of left exact morphisms

[Joyal, André]

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


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Tensor product of left exact morphisms

[Richard Garner]

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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Re: Tensor product of left exact morphisms

[henry]

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
>


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Re: Tensor product of left exact morphisms

[Richard Garner]

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
>
>

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