Monday, January 17, 2011

??almost seems like underlying finitely cocomplete algebroid of an abelian category obeys some extra "equational law" (or something...), but that's probably not correct... ??so then what _is_ going on?? ... ...??something about tensor product as well, or ... ???... ??hmm, what _about_ structure not preserved by the morphisms, but which might somehow affect the structure that _is_ preserved?? ... in general... ??doesn't it seem like this could happen, in general?? ... ??or something???... ??any conspicuous examples??? ...

??try to develop simplicial approach to rep of groupoid, then adapt to lie algebroid case??? ???or something?? ???something about... ???for each j, we should have a vector bundle over the set of j-simplexes in the nerve of the groupoid... with the fiber over a particular j-simplex x being the vector space of "coherent sections" of the representation over x ... ??or something ...

???hmm, so what about dg modules of a dgca as special (??in just what way???) simplicial modules of its dold-kan correspondent ?? ... or something...

??idea that "koszul duality" (or something...) in "lie algebra case" (or something...) might work a bit differently, or perhaps rather be capable of being thought of a bit differently, from the way that i was thinking about it ... ??i was thinking something like... "a lie algebra as a vector space together with some sort of alternative dgca structure on its exterior algebra ..." .... ????or something????...... ???whereas now it seems like it should be something like... ??"a lie algebra as a certain kind (meaning with just plain property...) of dgca" ... ???or something???

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