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 ___________"
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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
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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
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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...
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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))
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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
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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...
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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
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ugh this actually kinda forces me to think about a bunch of details i was sweeping under the rug ewwwww
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Replying to
if you're willing to only think about ∞-groupoids you can uhhh "simplify" the story somewhat by tossing out F and just asking what classifies "bundles of ∞-groupoids over X," the answer will literally just be maps from X into the ∞-groupoid of ∞-groupoids
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Replying to @schala163
lol imagine writing down cocycles instead of simply stating everything invariantly and then not calculating any examples
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also, "bundles of ∞-groupoids over X" just means maps into X, it's great stuff; in other words "fibration" isn't itself an invariant notion, every map is a (disjoint union of) fibration(s) from the homotopy pov
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right of course
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