??so is there some nice generalization of "zariski tangent space of an algebraic variety" where the variety is generalized to a stack and the tangent space is a chain complex or something??? ???

but then what about zariski tangent space vs tangent cone?? ... and so forth... ??...

??hmm, so what _about_ the idea of "a filtered algebraic-geometric theory" and so forth?? and how it relates to stuff like "algebraic stack of poisson algebras as normal stack to substack of commutative algebras inside ambient stack of associative algebras" ??? ...

hmm, something about the way that... a filtration on an operad (for example...) can cut down to a quotient operad (if it's the ideal power filtration of the corresponding operad ideal... i guess that we still have to get around to thinking about the more general case of a filtration that's not just an ideal power filtration...), but not more finely, to something really point-like... for that it seems like you have to go to a richer doctrine... ??hmm, does that really make sense??? ....

(what about "intended environment" for operad doctrine?? or something...)


??what about a normal cone (or something...) as sort of like "a tangent cone to a somewhat generic point" ?? ??or something???

anyway, so one thing that we should probably do is to try to work out how the whole big "blowing-up of a sub-stack" (or something...) story should reduce in some special case (or something) to some concept of "zariski tangent space of a point of an algebraic stack" (or something), which might be some sort of chain complex or something... ??...

we should really try, for example, taking the dimensional theory of "cubo-quadratic algebras" and considering on it the ideal power filtration of the ideal corresponding to gauss's lemniscate; and looking particularly closely at the first (and also zeroth??) associated grade... or something... ??...