@D.ZackGarza No, this is always true (essentially by definition of weak equivalence: we can probe things only with spheres so we don't see any of the pathologies)

There are however more refined invariants that do see the pathologies (e.g. the shape of the topological space)

Another basic question: Let $\xi$ be a vector bundle over a (good-enough) space $M$. Consider the fiberwise stabilization $E$ of the spherical fibration associated to $\xi$. Is it true that the Thom spectrum associated to $\xi$ is weak equivalent to the space of global sections of $E$?

Strictly speaking, I was not referring to the "space" of global sections, but a kind of $R\Gamma$, which has contributions of negative degrees. I don't know the correct terminology.

The thom spectrum is the colimit of the spherical fibration considered as a functor from the space to the ($infty$) category of spectra.

Yeah, and global sections are limits in nature.

I am trying to understand whether Atiyah duality could be understood as a kind of Verdier duality. If it were true, then $M^{-TM}$ would be global sections of a sheaf $f^!\mathbb S$ where $f\colon M\to\operatorname{pt}$ is a map from a smooth manifold $M$ to a point. I believe that this should be something already well-understood.

Yeah, so I think some version of what you're saying is definitely correct and I thought about this myself several times in the past. I'm not sure where this verdier duality for spectral sheaves is developed in the literature though. I'm also confused about what happens for a general map of manifolds. How does verdier duality work there... Hoping someone chimes in with a full answer.

@Yai0Phah a neighborhood of 4.3 in arxiv.org/pdf/1112.2203.pdf might be of interest. i’ve never read it to know if it’s exactly what you want, but it’s definitely of a similar flavor

Thanks for the reference. I saw Denis Nardin mentioned Rezk's notes about Atiyah duality. Given that this is something very classical, I suppose that Verdier duality should have been already well built, however my ability of googling is too weak to find the exact theory out.

Did you read Lurie's notes about Verdier duality? Seemingly he builds the thing in infty-categories. I am very unfamiliar with manifolds (PL- or smooth), so I cannot tell quickly whether his approach is strong enough to establish a spectral version.

@Yai0Phah His approach is definitely strong enough to establish a spectral version, and it is kind of something known by the experts, although I'm unsure whether it's been written down properly. I think Thom spectra ought to be homology, so something like $f_!f^!\mathbb{S}$ rather than $f_*$.

A lot of the stuff has been written down only for local systems, but constructible sheaves of spectra are definitely within reach of current technology, and at least a little bit has been explored

@Yai0Phah HA.5.5.5.13(i) should let you mimic the classical argument by reducing to the identification of the dualizing sheaf for R^n.

@DenisNardin Thanks! I was pretty sure this was the case, but wasn't sure if some condition like Hausdorff/2nd countable/etc was implicitly assumed somewhere.

I was a bit thrown off by this result, but I'm guessing "having the homotopy type of a CW complex" means something like "admits a (strong) homotopy equivalence to a CW complex"? arxiv.org/abs/math/0609665