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