i.e. it's murky whether there's a canonical way to record all this data.

On an unrelated note -- is it possible to have a Bousfield localization $L$ such that $L \Sigma^\infty \mathbb C \mathbb P^\infty$ is dualizable in $Sp_L$?

It seems unlikely, but then again a priori it would also seem unlikely that $L_{K(n)} \Sigma^\infty BC_p$ is dualizable, which it is.

@ThomasRot The reference I usually check for this is Bredon, Sheaf Theory. There is also Wilder, Topology of manifolds, but unfortunately I can't remember what's actually in there (other than a lot). The existence of an $L$-cohomology fundamental class for an $L$-cohomology manifold is Bredon's Thm 16.16.

Poincare/Alexander duality is discussed in chapter 9.

@TimCampion Is there a particular goal or expectation? It's hard to know what to try to say.

By the way, when I can write a clear resolution to my comments, I will do so. I need to focus on other things for now though; I would ignore them until then.

@MikeMiller Yeah, I guess that's the thing -- I feel as though the "directedness" of an $\infty$-category should mesh well with the directionality of spacetime, but I don't have a goal more specific than trying to fit those two things together. There's probably some relevant literature I could take a look at.

And it seems like the starting point -- writing down whatever categorical construction this is -- really begs the question of what level of geometric structure should be considered.

Well, I'm content to just use the piecewise smooth things, I'm not so worried about the continuous paths. (Though, in thinking about your construction, the structure you're introducing seems to essentially be a germ of a local light cone near every point, and as you observe you can straightforwardly extend that notion to continuous curves.)

@TimCampion i don't think L can come from a ring spectrum R. the periodicity theorems say that R ^ K(n) is non-null for some K(n), and in fact equal to a nonempty wedge of K(n)s, so that there's a factorization L_R –> L_(R ^ K(n)) = L_K(n) of the units for L_R and L_K(n). if L_R CP^∞ were already dualizable, then i think this factorization entails that L_K(n) CP^∞ would dualizable too, which is not true for any choice of K(n)

still nice to know something concrete

Thanks, that's a great point!

@TimCampion, @EricPeterson it seems possible to me that one could show that any localization $L_E$ that does not have the property that $\langle E\rangle\leq \langle HF_p\rangle$ must have the property that for some $n$, $\langle E\rangle\geq \langle K(n)\rangle$. I.e. you'd be getting that factorization whether or not you've got a ring spectrum. I think the main idea here is the minimality of $\langle K(n)\rangle$

Actually I think I retract my statement... because this actually kind of depends on the telescope conjecture.

If the telescope conjecture fails then there are these spectra $A(n)$ which are the fibers of $T(n)\to L_{K(n)}T(n)$, are also minimal.

@JonathanBeardsley How would the argument go, assuming the telescope conjecture?

Like, even if $\langle K(n) \rangle$ is minimal, I've never quite understood how close $\vee_n \langle K(n) \rangle$ is to $\langle \mathbb S_{(p)}\rangle$ -- it's not as simple as $H\mathbb F_p \vee \vee_n \langle K(n) \rangle = \langle \mathbb S_{(p)}\rangle$, is it?

Does every nonzero Bousfield class have a minimal one below it?

Is there a name for "the Bousfield class of a ring spectrum", or a characterization? They seem pretty special. For instance, there seems to be a sort of "infinitary smash product" on them -- you can take the colimit of finite smash products along unit maps. As a result, dual Zorn's lemma applies and it follows that every Bousfield class of a ring spectrum lies above a minimal such.

(I suppose you need to know Ohkawa's theorem to apply Zorn's lemma)

They're also idempotent under smash product.

no idea. i did want to highlight that the ring spectrum part of the above argument is only there to guarantee an n with R ^ K(n) not null, but the same tail of the argument would also say that any putative E with L_E CP^∞ dualizable must have E ^ K(n) null for all n, which sounds like a frustrating class of spectra to me

for sure

@TimCampion Yeah $\vee\langle K(n)\rangle\vee\langle HF_p\rangle$ is quite far from $\langle S_{(p)}\rangle$ if I recall correctly.

It's not even $\langle BP\rangle$

My idea ultimately was that every Bousfield class has the property that $\langle E\rangle\geq \langle 0\rangle$, and so either its minimal (i.e. there is nothing between it and $\langle E\rangle$) or it factors through a minimal Bousfield class.

Then, I was thinking that there was a classification of the minimal Bousfield classes away from $\langle HF_p\rangle$, but now I'm not so sure that's true.

@TimCampion I'm also not familiar with any special name for the Bousfield class of a ring spectrum. They are not unique in being idempotent.

One thing that seems to provide a lot of counter-examples is the Bousfield class of the Brown-Comenetz dual of the sphere, $\langle I\rangle$, so it might be worth seeing if you can figure out (or if someone else knows) what $L_I(\mathbb{C}P^\infty)$ looks like.

It's conjectured that $\langle I\rangle$ is minimal as well. Also, if the $A(n)$ spectra are non-trivial (i.e. the telescope conjecture fails) then their Bousfield classes are also minimal.

So anyway, I think this is all just getting us back to Eric's declaration that weird stuff happens in the Bousfield lattice.

My intuitive picture of the Bousfield lattice is that it has minima at all the $\langle K(n)\rangle$, and then it has this kind of "trap door" at $\langle HF_p\rangle$ below which all hell breaks loose (and this is where $\langle I\rangle$ lives).

Thanks

@JonathanBeardsley But is it actually true that every Bousfield class lies over a minimal class? I can see that it must lie under a maximal class, since sups are easy to understand, but it seems like in principle you could have a decreasing series of nonzero classes whose inf is zero

Like there could be part of the Bousfield lattice which looks like the lattice of cofinite-or-empty subsets of $\mathbb N$ or something -- no minimal classes.

@TimCampion they're pretty different: the brown-comenetz dualizing spectrum is not killed by the sphere but is killed by everything in that giant wedge. btw for these sort of questions you should check out "The structure of the Bousfield lattice" by hovey and palmieri