Re: Re: sheaves on localic groupoids

[Christopher Townsend <info@christophertownsend.org>]

Hi Eduardo & all

If F is a Grothendieck topos then there is a bounded geometric morphism p:F->Set. Let B be a bound for p, then there is a locale of surjections [N->>B],  where N is the natural numbers. This is a locale over F that can be mapped to a locale over Set (apply the direct image of p to the frame of opens of [N->>B]). You get the object locale of the localic groupoid that ‘represents’ F via Joyal and Tierney. The morphism locale is the image of  [N->>B]x[N->>B]. 

Not sure if that sheds much light on what the localic groupoid really is, but it is one way of constructing it. 

I’d like to add that any geometric morphism p:F->E, we know, gives rise to a ‘localic’  adjunction between locales over F and locales over E; the right adjoint being effectively pullback in the category of toposes. If you can find a locale W over F such that W->1 is an effective descent morphism and the slice of this localic adjunction at W is an equivalence then in fact the adjunction is a connected components adjunction of a localic groupoid in E; this can be shown by application of Janelidze’s  categorical Galois Theorem (the result holds at the generality of cartesian  categories - it’s quite straightforward). If you know that the localic adjunction is a connected components adjunction it is easy to get the Joyal and Tierney result by restricting to local homeomorphisms.

But of course you should complain that it must be hard to show that any localic adjunction, sliced at W, is an equivalence. In fact even at the general level of cartesian categories it is not that hard once we recall that the localic adjunction is stably Frobenius (something that is in the original Joyal  and Tierney paper but not dwelt on). For a Frobenius adjunction to be an equivalence it only needs to have its left adjoint preserve 1 and its unit to be a regular monomorphism. At the slice, 1 is automatically preserved, so we are just left checking that the (sliced) unit is a regular monomorphism. It turns out that this is so for W=[N->>B] precisely when B is a bound, effectively completing a proof of the Joyal and Tierney representation theorem.

I hope it is OK that I’ve used this thread to effectively advertise some work(*) that I did a few years ago.

Kind regards, Christopher

(*) A localic proof of the localic groupoid representation of Grothendieck toposes Proc. Amer. Math. Soc. 142 (2014), 859-866

> 19 Oct 2018, at 19:57, Eduardo J. Dubuc <edubuc@dm.uba.ar> wrote:
> 
> 
> A time ago when I was working on the subject I was also very curious
> about the same (or related) questions of Johon Baez.
> More concretely, if you take a pointless connected atomic topos (we know
> they are):
> 
> How or which is the localic point that Joyal-Tierney using change of
> base take out of the hat ?
> 
> How or which is the localic groupoid of Joyal-Tierney ?
> 
>> On 17/10/18 23:12, John Baez wrote:
>> Dear Categorists -
>> 
>> Joyal and Tierney proved that any Grothendieck topos is equivalent to the
>> category of sheaves on a localic groupoid.   I gather that we can take this
>> localic groupoid to have a single object iff the Grothendieck topos is
>> connected, atomic, and has a point.     In this case the topos can also be
>> seen as the category of continuous actions of a localic group on (discrete)
>> sets.
>> 
>> I'm curious about how these three conditions combine to get the job done.
>> So suppose G is a localic groupoid.
>> 
>> Under which conditions is the category of sheaves on G a connected
>> Grothendieck topos?
>> 
>> Under which conditions is the category of  sheaves on G an atomic
>> Grothendieck topos?
>> 
>> Under which conditions is the category of sheaves on G a Grothendieck topos
>> with a point?
>> 
>> (Maybe we should interpret "with a point" as an extra structure on G rather
>> than a mere extra property; I don't know how much this matters.)
>> 
>> Best,
>> jb
>> 
>> 

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