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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