When studying manifolds using stable homotopy theory, it seems natural to not just consider the suspension spectrum of a manifold, but also the thom spectrum of the tangent bundle. However, as far as I can tell, the thom of the tangent bundle is no longer functorial in maps of manifolds, is it? Is there a way around this?

@TomBachmann Why isn't it functorial in maps of manifolds anymore?

Isn't it a composition of functors?

Is this because it fails to be homotopy-invariant or something?

Well if V -> W is a morphism of vector bundles, is there always an induced map of Thom spaces? I was thinking that V/V\0 -> W/W\0 will not make sense if the map is not injective.

Ah, I see, thanks

I might be confused about everything, but I think you can compute the thom spectrum as the colim in spectra of M -> BO(n) -> BAut(S^n) -> Sp or something like that. If that is case, it might be easier to understand the functoriality through this.

Suppose I have a map of bisimplicial sets X_.,. -> Y_.,. such that X_n,. -> Y_n,. is a weak equivalence for every n. Are there any conditions that imply the map of simplicial sets in level 0 X_.,0 -> Y_.,0 is then a weak equivalence?

(Actually in my case both X and Y are essentially constant in the sense that the structure maps X_n,. -> X_m,. are all weak equivalences.)

@ShayBenMoshe I think the problem is that you don't just need a map of manifolds, you need a map of manifolds which is compatible with the classifying maps of the tangent bundles. So if you have an arbitrary map of manifolds $M\to N$ then you've got two classifying maps $T_M:M\to BO(n)$ and $T_N:N\to BO(n)$ but it's not clear to me that you'll automatically get a commutative triangle over $BO(n)$ (or $BGL_1(\mathbb{S})$).

(I guess I'm even assuming they're of the same dimension there)

@RuneHaugseng arxiv.org/abs/math/0607820 may have what you are looking for. Joyal and Tierney spend a lot of time comparing the two ways in which one may view a bisimplicial set as a simplicial simplicial set. For example if your bisimplicial sets are (Reedy fibrant) complete Segal spaces, then X_.,k -> Y_.,k is a weak equivalence in the Joyal model structure for all k.

Since your bisimplicial sets are essentially constant, you would in fact only need to Reedy fibrancy.

Right, but unfortunately my bisimplicial spaces are not even levelwise Kan

Oh rats :/

@JonathanBeardsley I believe you are right. Here's the kind of functoriality I think you have. We can define the category Vect whose objects are pairs of a manifold and a vb on it (something like the Grothendieck construction of the functor Vect: Mfld -> Cat). And I believe that you have Thom functor Th: Vect -> Sp. Does that sound right?