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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

· Jun 18, 2020

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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QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Jul 2, 2020

hm you don’t want to be too quick to think of groups as “basically just groupoids,” something important happens during the passage from one to the other and it’s that you forget the basepoint of BG

chloe!

@schala163

Jul 2, 2020

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...

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Replying to

@schala163

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

12:55 AM · Jul 2, 2020Twitter for iPad

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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...

QC (in Berlin 7/29-??? w/ COVID

)

@QiaochuYuan

Jul 2, 2020

yeah you want BG to be unpointed there

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