Yes I do. Thanks, I will check out Lurie's book.

And I'm not sure I agree that the periodicity theorem "comes from" that stratification. But if you can ascertain that, then you're right, we shouldn't have something analogous in rings necessarily, unless we can find a similar structure.

What I mean is, there are analogues of the Morava K-theories, which are basically the residue fields $k(\mathfrak{p})$. These things homology acyclics (use Tor), in a sensible way, and seem to give a sensible thick subcategory theorem. The question for me is, can I find a self-map $f$ of a perfect complex $X$ of type $\mathfrak{p}$ whose cofiber is type (and this isn't really defined) $\mathfrak{p}+1$, but I'd like it to be some kind of minimal deformation of $\mathfrak{p}$ or something. Whatever that means. And I would like for this to induce periodic families in $[R,R]$.

Oh not to mention, is there any way to know whether or not your stabilized objects give some kind of "cohomology "theories" on your original category?

Man David this is great, thanks! There's a lot to digest, but it sounds like it's pretty much exactly what I'm looking for. In fact, I'll probably need a cellular hypothesis in anything I'm thinking about anyway, so that's fine. When you say that $M$ is a $D$-model category, what does that mean? Somehow tensored over $D$?

Thanks @Nick! I too am thinking about the Boolean algebra of localization functors! But probably if Bousfield didn't do much more with it, neither will I.

For some reason math stopped showing up after I wrote that last sentence, but $ aT(n)$ should be $\bigvee_{m<n}T(m)\vee F(n+1)$.

Aha of course, that's clever! And yeah, I guess it is equivalent, by precisely your argument above. Thanks again. :-)

There's also the notion in Boavida and Weiss' paper that linearization is a sort of sheafification: arxiv.org/pdf/1202.1305.pdf, which kind of reminds me a stacks! :)

Hi @Akhil, thanks! However, the statement that the category of cohomology theories is idempotent complete seems to be equivalent to saying that the category of spectra is idempotent complete?

Awesome! Thanks so much. I figured it must be true. That's immensely helpful (since I can apply to recent work on idempotent complete triangulated categories!).