talking to chris rogers about my recent realizations about mistakes that i'd been making in thinking about d-modules...

??something about ... ?connecting "dg scheme as algebraic stack" with "smfcca as algebraic stack" by figuring out what the "coherent sheaves" over the stack corresponding to a dg scheme are... in the case where the dg scheme is "codiscrete" (the de rham case), these coherent (or i guess quasicoherent actually... or something...) sheaves should be the d-modules... in more general cases it should be interesting and not so difficult to figure out what they are....

??relationship to dg modules of dgcas ??? .....

??does a d-module have some sort of "nerve" or "co-nerve" or something??? ....

and so forth .... ????....

hmm, at this point perhaps the questions we're asking making sense only for those dgcas that are sufficiently "low-dimensional" in a certain sense... i forget to what extent we might have previously worked out what this sense is... ??maybe just something about being generated by generators (and/or relators??? or something???) in low degrees???? or something??... ??hmm, might this give some clues about special dg modules that might correspond to dg modules?? or something??... and so forth...

???something about ... ??"koszul duality" ... "bar construction" ... "chevalley-eilenberg" .... ???? and so forth... ???....

hmm, seems like... might be that with certain "degree constraint" (or something) on dgca, we might be getting basically just stacks corresponding to action groupoids of lie algebra actions on commutative algebras ... ??or something??...
hmm, or maybe just something about groupoids in general ... or something... or i guess that i mean lie algebroids or something... ???....

??something about... ??thinking of a certain degree-constrained sort of dgca as a "lie algebroid", and describing what the "coherent sheaves" should be in terms of the dgca ... ???or something???...

what about ... ??allowing only lower-degree generators, but automatically _imposing_ higher-degree relators?? ... or something ... ????..... ??anything sesqui-clever here?? or something???....

??semi-direct product symbol "x|" ...???

weyl algebra s(v*) x| s(v) ...

induced morphism to s(v*) x| env(polynomial vector fields on v) ...

??but confusion about how it seems like there's almost a morphism going the other way... ??or something??? ... at least, that enveloping algebra maps into the weyl algebra... ???or something????.... ...and so forth...

??something about confusion between "covariant differentiation" and "lie differentiation" ... ???"lie differentiation" as something about s(v*) x| env(polynomial vector fields on v)... "covariant differentiation" as something about s(v*) x| s(v) ... ??or something???.....

maybe i should try to check how standard my usage of "semi-direct product" is here... ??not clear to me offhand how it relates to semi-direct product of group acting on other group ... ??... ??funny how in the case of the weyl algebra there's this sort of symmetry between the actor and the acted-upon ... ??or something??? ... also something about relationships between nilpotent and solvable ... ??and so forth???....

??vague memory of ...???being surprised by some sort of semi-direct product asepct (??in group sense maybe??) of weyl algebra... ?? ...??semi-recently?? ... ??what was_that about??... ??how closely did it tie in with old idea about quantum mechanics and semi-direct product and "measurement process disturbing the quantity being measured"?? ... and so forth... ??might it have been based on incorrect belief that i've recently been trying to recover from??? .... ??again of course, what about relationship to "semi-direct product of hopf algebra acting on commutative algebra" ?? ... and so forth... ???...

??questions (or something...) for chris rogers:

what exactly is a "lie algebroid"?? ... (not worrying about "lie n-algebroid" yet...)

can we think of lie algebroids as forming a reflective (or something) full subcategory of "dg manifolds" or something like that? (allow variations on "manifold" here of course...) exactly how?? some sort of "degree cutoff" or something??...

do we have a clear concept of "quasicoherent sheaf over a lie algebroid", so to speak?? (mainly in "algebraic" case, perhaps?? ... though maybe not exclusively ...) ???... if so, then can we think of these as something like special dg modules of the alleged corresponding dgca?? ... ??...

...and so forth...

??what about trying to relate diff eq aspect of d-module to that of dg module and/or of dgca ??? .... and so forth... ???....

??so what _about_ diff eqs involving infinitesimal simplexes... ??? ... ??"pfaffian..." .... ????....