On the one hand, we have BG the space, or the BG the “delooping” one-object groupoid with morphisms G, which we might write G => *. These are closely related; the former is the geometric realization of the nerve of the latter.

On the other hand we have BG the classifying topos or stack, which is (I think) the category of all principal G-bundles.

The notation and similar role played by those objects suggest they are versions of the same thing. On nLab, we find in classifying topos it reads says that the correspondence of toposes GBund(X) = Topos(Sh(X), G) is analogous to the correspondence pi0 GBund(X) = pi Top(X,BG).

Ok but is it just an analogy, or is there some kind of stackification or Yoneda process that turns BG the space/groupoid into BG the topos/stack? Or is there some kind of truncation or geometric realization process that turns BG the stack back into BG the space?

In the article moduli stack it says the moduli stack *//G is the base of the universal principal bundle. Does that mean in the category of stacks? What’s the “total space” over *//G? Is it a stackified version of EG?

It’s hard to believe that these theories are so utterly parallel just by coincidence without a literal connection.

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