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I had begun writing classifying topos of a localic groupoid and sheaves on a simplicial topological space, but am (naturally) being distracted from nLab work now, so this is left in somewhat unfinished form…
The nomenclature “classifying topos of a localic groupoid”, although present in the title of that cited paper of Ieke’s, sounds just a tad odd to my ears. Or is it usual to refer to $Set^G$ as the classifying topos of a group $G$?
Yes, we can think of $Set^G$ as topos-theoretic cousin to the classifying space $B G$, but there is another cousin on the topos-theoretic side, namely $Sh(B G)$. They are a bit different, because for nice spaces $X$ the geometric morphisms $Sh(X) \to Sh(B G)$ correspond to continuous maps $X \to B G$, whereas geometric morphisms $Sh(X) \to Set^G$ correspond to $G$-torsors in $Sh(X)$, or equivalently to homotopy classes of continuous maps $X \to B G$.
I don’t know whether it’s common to call $Set^G$ the “classifying topos of G.” It seems to me that tradition aside, it has as much right to be called that as the space BG does to be called the “classifying space of G” – in both cases what is actually being classified are G-torsors. But perhaps for that reason “the classifying space of G” was a confusing term from the beginning and we shouldn’t compound the problem.
Do I remember correctly that $Set^G$ and $Sh(BG)$ are actually “weak homotopy equivalent” as toposes?
Do I remember correctly that $Set^G$ and $Sh(B G)$ are actually “weak homotopy equivalent” as toposes?
Yes. This is discussed and proven in Ieke’s Springer Lecture notes.
It’s theorem 1.1, proven from page 77 on.
I have
expanded a tad at classifying topos of a localic groupoid,
made classifying topos of a topological groupoid a redirect to it,
added to Grothendieck topos a brief remark under Properties – As localic groupoids (since any such mentioning had been missing there).
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