Re: sheaves on localic groupoids

[<henry@phare.normalesup.org>]

Dear John,

A topos has a point if and only if it can be represented by a (?tale
complete open) groupoid whose space of objects has a point.
But it is also easy to construct pointless groupoids representing the
topos of sets: for any non zero locale X, the trivial groupoid on $X$
represents the topos of sets.

An explicit description of points of the topos attached to an ?tale
complete open localic groupoid as some sort of G-torsor can be extracted
from Ieke Moerdijk's paper the classyfing topos of a continuous groupoid"
(I and/or II), I don't think you can get anything better than that.

For the other property:

If $T$ has a groupoid representation $G$ then the coequalizer of $G$ in
the category of locales is the localic reflection of T. This follows from
the fact that sheaves over groupoids are colimits in the category of
toposes and localic reflection preserves colimits.

So any property of a topos that can be tested on its localic reflection
can be tested on this quotient. I believe this applies to connectedness as
the inverse image of the localic reflection geometric morphisms T -> L is
always fully faithful.


"Atomic" and "Connected atomic" have nice characterization as soon as one
restricts to open ?tale complete localic groupoids.

It is proved in sketches of an elephant C.3.5.14 that a topos T is atomic
if and only if both the geometric morphisms T -> 1 and T->T \times T are
open.
It also follows from combining several results of section C.3.5 that T is
atomic connected if and only if these two maps are open surjections.


Now if T is represented by an open ?tale complete localic groupoid G, it
means that you have an open surjection G_0 -> T and that G_1 = G_0
\times_T G_0

In particular by manipulating a little all the pullback square involved,
and using the fact that pullback along an open surjection preserve and
detect open map and open surjection (C5.1.7 in sketches) one can see that
T->1 is open (resp. an open surjection) if and only if G_0 ->1 is, and T
-> T \times T is open (resp. an open surjection) if and only if G_1 -> G_0
\times G_0 is.



Best wishes,
Simon Henry


> 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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