okay so I think I have a satisfying answer to this now...
the bit about BG being a homotopy quotient is a bit of a red herring. If we are thinking in terms of â-groupoids, then BG simply *is* G but viewed as a(n) (â)-groupoid
1/k i guess
idk how long this is gonna go
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related question: what is the "best" way to see the fact that the homotopy quotient of a point by G is the classifying space for principal G-bundles.
By "best" I mean something like... formally in the language of â-groupoids......
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