Let $X$ be a space (CW complex) and let $E \to X$ be a vector bundle.

Using the language of $\infty$-categories we can can define the Thom space $T(E)$ as the pointed space representing the following **$\infty$-pre-cosheaf** on the category of pointed spaces:

$$S \mapsto fib(Map(E,S) \to Map(E \setminus X, S))$$

In words, a a pointed map from $T(E)$ to any pointed space $S$ is given by the data of a map $E \to S$ together with a null-homotopy of it's restriction to the complement of the zero section $E \setminus X$.

This is all very nice however most of the interesting occurrences of Thom spaces are when they are mapped into rather than from. The raw imprecise question is therefore the following:

Is there an abstract definition of some manifestly interesting $\infty$-presheaf which $Map(-,T(E))$ happens to represent?

Here's what makes me hopeful:

In Ranicki's book on surgery he proves (what I think is) the following statement about Thom spaces:

Let $X$ be a manifold, $\eta$ a rank $k$ bundle and $N$ an $n$ dimensional smooth manifold.

**Definition:** The Bordism set $\mathfrak{B}_{m}(N,X, \eta)$ is defined to be the following set of equivalence classes:

**Elements** are $m$-dimensional submanifolds $j:M \hookrightarrow N$ ($m=n-k$) together with a map $f: M \to X$ s.t. $f^* \eta = \nu_{M \hookrightarrow N}$.

Two such elements $(M_1,f_1)$ and $(M_2,f_2)$ are **equivalent** iff there exists a submanifold $i: W \hookrightarrow N \times I$ and a map $F: W \to X$ s.t.

$\partial W = M \amalg N$

$F^* \eta =\nu_{W \hookrightarrow N \times I}$

$F |_{\partial W} = f_1 \amalg f_2$

Theorem:Transversal intersection with the zero section induces a natural bijection:$$\mathfrak{B}_m(N,X,\eta) \cong [N,T(\eta)]$$

This statement seems to hint at the possibility of defining for every vector bundle $E$ on a manifold $X$ the Thom space as a presheaf on the category of smooth manifolds, whose $\pi_0$-presheaf is $\mathfrak{B}_{m}(-,X,E)$. Here's the question (which we have already answered the 0-categorical version of).

Question:Is there a categorical (bordism-flavoured) definition for the $\infty$-presheaf on the full subcategory of spaces generated by spaces homotopy equivalent to smooth manifolds which assigns to every manifold $M$ the mapping space $Map(M,T(E))$? (as we saw the $0$-categorical version is solved by $\pi_0(?)(-)=\mathfrak{B}_m(-,X,E)$)

Hopefully once the above is settled we could define the Thom space as the space representing the left-kan extension of the above presheaf along the inclusion functor to the category of spaces.

Elementary proofs of some results of cobordism theory using Steenrod operations(he does the complex-oriented case, but it shouldn't be too hard to do the general case along the same lines). I don't know if this can be extended to give a description of the mapping space though. $\endgroup$ – Denis Nardin Jan 5 '18 at 21:15