it’s like uhhh you take a lie groupoid and differentiate it right
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yeah okay ummm...
what? does that means exactly??
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okay so a lie groupoid is like a smooth manifold together with a bunch of “smooth isomorphisms” between points right? (thinking about a smooth action of a lie group helps here maybe)
you differentiate that and get infinitesimal isomorphisms between infinitesimally close points
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this is the vector bundle + the anchor map in the anchor map definition; the vector bundle is “infinitesimal isomorphisms” and the map to the tangent bundle tells you which infinitesimally close points are being identified
then the lie bracket encodes composition
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okay so im mostly confused about the lie bracket part... ummmm
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so uhh what is a lie bracket?
i think of it like
derivation (of a bilinear form) = the kind of thing that you can formally exponentiate to get an automorphism of that bilinear form
the jacobi identity says that [X,-] acts via derivations on the lie algebra itself...
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yep, and that exponentiates to the adjoint action of G on g
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umm okay so a lie algebroid consists of
-a vector bundle together with a lie algebra structure on its space of sections
plus some other stuff that im not gonna think about rn
this should be the lie algebra of some lie group?
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like if i have the tangent lie algebroid on X, then the global sections give me the lie algebra of Diffeo(X), right?
umm "lie group" should be quotes there cuz it's some infinite dimensional thing but like... yah
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if i have a lie groupoid is there some way of getting a (possibly infinite dimensional) lie group out of it?
where if i start with the smooth path groupoid i get back Diffeo(X)??
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this construction makes me antsy, i have the sense that it's a little unnatural but i can't quite justify it 🤔
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well im not sure id call it a "construction"... it's a bit under-specified
im going lie groupoid -(differentiation, a functor) -> lie algebroid -(global sections, also a functor)-> lie algebra -(integrate, not actually a well-defined operation) "lie group"
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but it's suggestive enough that i feel like there ought to be a more direct construction to go from the lie groupoid to a lie group
otoh, i agree that this seems weird as heck
tbh im kinda just poking at the definition and saying "this seems weird"
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