(the object $E_0$ over $X^{\times 0} \simeq pt$ will be required to be the sphere spectrum $S$, and over $X \xrightarrow{p} pt$ you associate the unit map $p^*(E_0) \to E_1$)
...what are you trying to do with all of this, anyways?
by the way, all of this was relative to the "external tensor product" symmetric monoidal structure on $T_S$ that i think you were describing above. you can also make $T_S$ into a bundle of symmetric monoidal categories whose fibers have the pointwise tensor product (though the total category is no longer symmetric monoidal)
@JonathanBeardsley sorry, i made a reduction that i wasn't being clear about. if you're looking for the operad $Fun(X,Sp)^{\otimes,ptwise}$ inside some other category, a good first step is to talk about what the global sections should be, i.e. the commutative algebras -- and that's what i was describing. the operad itself will be a "relative" version of this, i.e. $Fun^{rel}_{/Fin_\ast}(Fin_\ast,ABC)$ (where "ABC" is the unknown thing, which lives over $Fin_\ast$) or some subcategory of that
oops, the source should probably be doubly-pointed finite sets or something like that
Given a commutative ring spectrum $R$ and an $R$-algebra $S$, under what conditions do we have something like $S\otimes_RM\simeq0$ implies $M\simeq0$ for any $R$-mod $M$?
in analogy to classical algebra one might hope that R -> S being faithfully flat suffices, but an additional finiteness condition is necessary: pi_0(B) should be generated by at most ℵk generators and relations over pi_0(A)
more generally, the desired statement is true if R -> S is "descendable", i.e., that Tot of the cobar complex of the map R -> S in commutative R-algebras is R
you should check out this paper: arxiv.org/abs/1404.2156. (btw in my first message, B should be S and A should be R)
for instance, the map KO -> KU satisfies the desired property, and is not flat. I also garbled the definition of descendability: the tot tower of the cobar complex should be pro-constant
@FrankScience This is almost certainly not the answer you want, but for $M$ perfect this is equivalent to $\mathrm{Spc}\,S→\mathrm{Spc}\,R$ being surjective on closed points, where $\mathrm{Spc}$ is the Balmer spectrum of the ∞-category of perfect complexes
Of course the problem is that computing $\mathrm{Spc}\,R$ is basically impossible but one can dream...
Here's a question about Dotto's equivariant Goodwillie tower -- or maybe it should be the "equivariant Goodwillie castle" since there are a bunch of towers sitting inside it! The question is: do the layers of the the castle for the identity functor form a genuine equivariant operad?
@TimCampion What kind of objects the layers are is, as far as I know, still an open question. It is certainly hoped that something like what you said is true
In his paper, there's also an issue where he's only able to classify "strongly homogeneous" functors, rather than all homogeneous ones. I suppose that's an additional complication.
It was hoped for a while that it was exactly the kind of spaces that form the objects of an equivariant operad (spaces "genuine wrt graph subgroups of G×Σ_n"), but Emanuele told me he had a counterexample
Sarah Yeakel's theorem that taking derivatives is a lax monoidal functor from functors to symmetric sequences gives a conceptual reason why the ordinary derivatives of the ordinary identity form an operad. But I guess you're saying it's already unclear what kind of object the derivatives are, so it's not even clear how to correctly think of "taking derivatives" as a functor.
Does anyone know if there's an Eilenberg-Watts theorem for operads? E.g. given a functor from O-algebras to P-algebras with such and such properties, we can identify it as tensoring with blah. Would be fine with something in characteristic zero.
