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
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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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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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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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yeah you want BG to be unpointed there
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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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I might need to think a bit about what "pointed" even *means* when I'm talking about ∞-groupoids...
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yeah it's a good exercise to work out explicitly the thing i said earlier about the difference between groups and connected groupoids; explicitly, to write down what a natural transformation between two functors BG -> BH is, where G and H are discrete groups
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I've worked that out before (though I should prolly do it again...)
but I meant something more basic like umm
I guess I'm worried that just talking about the "point" in "pointed" might be a bit evil? like who am I to pick out this one contractible space and treat it as special
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nah it's fine, there's a contractible space of contractible spaces
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also it's the terminal object in ∞-groupoids ("the" meaning, as usual, up to a contractible space of choices)
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"but if I picked some other large contractible space and used that instead then that would be weird... like 'spaces equipped with a map from S^∞' seems very different from pointed spaces..." - me
also me: DON'T MAKE ME TAP THE SIGN
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i feel like "homotopy theory isn't about spaces" is maybe gonna be my "thing" for a while lmao
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