so, based on my (super limited!) understanding, these kinds of things should appear as the "coordinate rings" of some kind of derived spaces (here spaces could mean manifolds or schemes or something else, i wanna be vague)
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think about the de rham complex of X for a second (this is the chevalley-eilenberg algebra associated to the lie algebra of vector fields)
as a co-simplicial ring, i think this is something like "functions on the space of tuples of infinitesimally close points"
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you can start with X and form a simplicial object that looks kinda like the singular simplicial set of a space, except you only use "infinitesimal simplices"
without saying anything substantial about derived stuff (cuz i dont know how >__<), this is a simplicial space, so...
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... it's "coordinate ring" should be a co-simplicial commutative algebra.
And if I'm not totally lost, this spits out exactly the derham complex.
I'm basing this mostly off my vague memory of synthetic differential geometry, where k-forms are basically *defined* this way...
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now if g is a lie algebra, i am wondering if there is some nice way of viewing its the chevalley-eilenberg algebra as "functions" on some kind of "derived/simplicial" space in the same way
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I have a guess (mostly just a shot in the dark tbh): g encodes something like a "formal nhbd of e" of some lie group
imo the answer should be "infinitesimal simplices" where the basepoint gets sent to the e. Because ummm that's the first simplicial object i could think of lmao
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okay so ive been told that 1) this guess is basically correct (think i made an off-by-one
but more importantly
2) here is an alternative description: One way of making the lie algebras <-> formal groups thing precise (thread incoming) has as an upshot that lie algebras are
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equivalent to group objects in an appropriate category of "infinitesimal spaces" (short version: formal group k-schemes where e is the only point, or something close to that)
if you have a group object, you can try to deloop it by doing a bar construction!
2/3 oops
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This will spit out a simplicial object (that u should be able to interpret in any reasonable category of derived spaces by taking a homotopy colimit... nvm that) which deserves to be called Bg... and it's *almost* the thing I guessed at >__<)
3/4 classic chloe
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so the upshot of this all is: the chevalley-eilenberg complex of a lie algebra g is the coordinate ring of the classifying space of the formal group corresponding to g!
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whoops, wrote a comment but i see you already got there. yes, that’s right. this relationship between lie algebras and commutative algebras is a form of koszul duality, which is secretly about looping and delooping
DAG X has some good stuff on this (dg lie algebras as a model for formal stacks): people.math.harvard.edu/~lurie/papers/
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koszul duality also gives the relationship between the quillen (lie algebra) vs. sullivan (commutative algebra) approaches to rational homotopy theory. lots of pretty stuff, and the rational homotopy version is explicit enough to do explicit calculations with
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