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· 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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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

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...

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

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...

yeah you want BG to be unpointed there

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??

oof okay i need to be careful here. BG does have a distinguished basepoint, and as you say it does correspond to the trivial G-bundle. when people say "maps X -> BG classify G-bundles" what they mean invariantly is that there's a space (oo-groupoid) of G-bundles on X and it...