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chloe!
@schala163
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 ∞-groupoids, then BG simply *is* G but viewed as a(n) (∞)-groupoid 1/k i guess idk how long this is gonna go
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chloe!
@schala163
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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 ∞-groupoids......
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chloe!
@schala163
hmm i prolly have to think about that more... but i don't *think* it's gonna screw me up for this thread at least...
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QC (in Berlin 7/29-??? w/ COVID 😷)
@QiaochuYuan
to elaborate, in the discrete case, there’s a 1-category of groups and an equivalent 1-category of pointed connected groupoids and a *not* equivalent 2-category of connected groupoids; the latter has nontrivial 2-morphisms given by pointwise conjugation
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chloe!
@schala163
mmm so im gonna wanna say stuff like "maps X->BG correspond to maps of ∞-groupoids". And really "corresponds" should mean "there's an equivalence of ∞-groupoids", but I guess it's gonna be really important to know whether I'm talking pointed or not when I try to say that...
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chloe!
@schala163
Replying to
@QiaochuYuan
hmm but the things I'm classifying with these *are* pointed right? cuz like taking the trivial fibration should correspond to mapping to the basepoint in BG??
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