@AaronMazel-Gee @ArunDebray Somebody needs to write this article though!

Anyone know general conditions from, say, HA, under which operadic modules over a an $O$-algebra $A$ form a stable quasicategory?

In particular, I'd like to start with an $O$-algebra $A$ in $Ch(\mathbb{Z})$, so that's a pretty nice situation I think.

I think when the underlying category has a monoidal structure that is compatible with colimits then the forgetful functor from O-A-modules to underlying objects should preserve limits and colimits

maybe that's what's being said in HA.3.4.4.6?

@DylanWilson yeah that's def true

Yeah okay so I guess that gives it to you.

👍

I sort of feel like a really good thing to do in homotopy theory would be to figure out the Balmer spectra (or more generally the Bousfield lattices) of interesting ring spectra (i.e. their categories of modules)

But maybe that's too hard...

Like what are the points of Spec(S[x])? Or for interesting DGAs? Or TMF?

@SaalHardali do you mean "homeomorphism invariant" as in "only depends on homeomorphisms that aren't necessarily diffeomorphisms"? if so, i would actually hope not, since that'd lead to a richer invariant. a priori i don't see any reason why it should be invariant under arbitrary homeomorphisms, that sounds to me like it would be a nontrivial theorem

are there any interesting pairs of a smooth manifold $M$ and a natural number $k$ for which the space $Emb(M,\mathbb{R}^k)$ is tractable? for instance, even when $M$ is $S^1$ or $R^n$ this seems potentially nontrivial.

@JonathanBeardsley agreed. i would moreover urge whoever does write it to embrace the newer meaning of "homotopy theory", as promoted e.g. in clark's open letter

@AaronMazel-Gee I'm not saying this is trivial but I'm just saying it would be weird to imagine a case where being non-characteristic depends on the smooth structure...

@AaronMazel-Gee A good exercise, surely!

@AaronMazel-Gee For k>>dim(M) one can combine embedding calculus in M and orthogonal calculus in R^k to access the rational homotopy type of Emb(M,R^k), see e.g. work by Arone et al. For M=R^n, n<<k and requiring boundary conditions, this is what people call the higher-dimensional version of the space of long knots which is (rationally) reasonably understood as well.

@archipelago ah of course. thanks for reminding me of that whole field! ;o)

@JonathanBeardsley TMF is known at least, arxiv.org/pdf/1311.3940.pdf

@Drew oh nice!!

is there some geometric interpretation of ku(X) when X is, say, a smooth manifold or compact Hausdorff space or something?