Dear John, Eduardo and Simon, A few comments and questions. If G is a connected localic group, then every continuous action of G on a set is trivial. Hence the topos of G-sets contains no information about G. This is annoying. Of course, we could replace G-sets by G-locales (= actions of G on locales). But the category of G-locales is not a topos in general. Should we consider the gross-topos of sheaves on the category of G-locales ? What should be the Grothendieck topology? What are the applications ? (the applications may guide the developement of a theory). I do not have a satisfactory answer to these questions . Remark: There are plenty of connected Lie groups. Equivariant homotopy theory is an important branch of topology. Best wishes, André You are considering the topos T of equivariant G-sheaves on a localic groupoid G. In the case where G is a group, T is the topos of continuous G-sets. Under which conditions is the category of sheaves on G a connected Grothendieck topos? ________________________________________ From: John Baez [baez@math.ucr.edu] Sent: Wednesday, October 17, 2018 10:12 PM To: categories Subject: categories: sheaves on localic groupoids 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/ ]