Re: Tensor product of left exact morphisms

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

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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