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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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thinking about *locally trivial* bundles is also a bit of a distraction. If I just give you X as a homotopy type/∞-groupoids, you can't even say what a "locally trivial bundle over X" should *mean* Don't make me point to the sign: twitter.com/schala163/stat 2/k

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· Jul 1, 2020

i feel like "homotopy theory isn't about spaces" is maybe gonna be my "thing" for a while lmao

we should really just be talking about fibrations in general we might wanna be able to say something like "fibrations over X with (homotopy?) fiber F are classified by ___________" 3/k

here F could be all sorts of things. The easiest to think about is if F is just a set (or a really a setoid I guess). But we might also want to allow F to be a higher groupoid in it's own right... later 4/k

Also instead of being "just" a set(oid), F might have some extra structure, e.g. a vector space or some other algebraic gadget... for G-bundles, we are talking about fibrations of G-spaces where the (homotopy?) fiber is the left(or is right better?) action of G on itself

so I think that what should happen in general is that F-fibrations are classified by maps to the delooping of Aut(F) umm I'm a little unclear about whether I should be talking about pointed or unpointed stuff here... 6/k

if F is just a set(oid) then Aut(F) is just a group, and it's delooping is a groupoid, and this should recover the classification of (associated bundles of) Aut(F)-principal bundles by maps to B(Aut(F)) 7/k

if F is something fancier, say an n-groupoid... then I'm not quite sure but I *think* that makes Aut(F) an n-group and its delooping an (n+1)-groupoid... I definitely need to think about this more lmao 8/k

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

okay so why should this be true? well suppose we have a fibration F->E->X we need to get a map X->BAut(F) from this data where ofc by "map" I mean "morphism of ∞-groupoids"... I'm a bit confused cuz it *feels like* I want everything to be pointed, so maybe ∞-groups... 9/k

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@QiaochuYuan

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1:48 AM · Jul 2, 2020Twitter Web App

nah you want a map of ∞-groupoids, X doesn't need a basepoint here (and e.g. in the extreme case could be empty)

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hmmm so I guess I've kinda been keeping a basepoint around for X the whole time just because I like being able to say "F is the fiber over the base point" but like that's not really doing any work so yeah I should dump it alright cool