$\infty$-categorical description of $n$-manifolds

Question

$\newcommand{\Mfld}{\mathsf{Mfld}} \newcommand{\Space}{\mathsf{Space}} \newcommand{\Sh}{\operatorname{Sh}} \newcommand{\PSh}{\operatorname{PSh}}$ I am wondering there is (or is expected to be) an $\infty$-categorical description of the category $\Mfld_n$ of $n$-manifolds with morphisms given by the space of codimension zero open embeddings.

For example, here is one possible result in this direction. There is the fully faithful Yoneda embedding $\Mfld_n \hookrightarrow \PSh(\Mfld_n)$, presheaves of spaces on $\Mfld_n$. I also found the following result in arXiv:1409.0501 that the category of sheaves on $\Sh(\Mfld_n)$ is equivalent to $\Space_{BO(n)}$, spaces with a map to $BO(n)$ (this equivalence takes a space $E\rightarrow BO(n)$ to the sheaf $\phi_E$ with $\phi_E(M)$ equivalent to the space of sections of the pullback $E \times_\tau M$, where $\tau: M \rightarrow BO(n)$ is the tangent bundle classifier). Here, $\Sh(\Mfld_n)$ is the localization of $\PSh(Mfld_n)$ with respect to covering morphisms. That is, in $\Sh(\Mfld_n)$, we add isomorphisms between a smooth $n$-manifold, viewed as a colimit of open disks, and a formal colimit of open disks in $\PSh(\Mfld_n)$. So if we are allowed to chop up manifolds "freely", then this category is equivalent to $\Space_{BO(n)}$.

Also, arXiv:1206.5522 proves that for topological manifolds, factorization homology gives an equivalence between $\mathsf{Fun}^\mathrm{Ex}(\Mfld_n, C)$, symmetric monoidal excisive functors from $\Mfld_n$ to a symmetric monoidal category $C$, and $E_n\text{-}\mathsf{Alg}^C$, $E_n$-algebras valued in $C$. Excisive again means we are allowed to chop up our manifolds "freely."

Question: do we expect (or not) a $\infty$-categorical description of the category $\Mfld_n$, without allowing the "free" chopping up referred to above?

Ideally, such a description would involve some local data, like an $E_n$-algebra, but perhaps that would lead to choppping up $n$-manifolds. If there is no description involving local data, is there description of $\Mfld_n$ in terms of something more algebraic?

Answer 1

There is a specific "$\infty$-categorical" approximation of manifolds which your question seems to converge to. Before we address that, I would like to point out that Ayala-Francis do not classify excisive functors $(\mathrm{Mfld}^\mathrm{Top}_n,\bigsqcup) \rightarrow (C,\otimes)$ by $E_n$-algebras, but rather they classify them by algebras over the topological, framed $E_n$ operad: $E^{\mathrm{BTop}}_n$. This operad is composed of the topological embeddings $\mathrm{Emb}^\mathrm{Top}(\bigsqcup_i \mathbb{R}^n, \mathbb{R}^n)$. This is not a purely $\infty$-categorical object because we know of no description of this operad which doesn't use the topology of $\mathbb{R}^n$. If one instead passes to smooth manifolds things get better, but one is not completely out of the woods. Ayala-Francis classify excisive functors out of this category via the framed $E_n$ operad: $E^\mathrm{fr}_n$. This is the smooth embeddings $\mathrm{Emb}(\bigsqcup_i \mathbb{R}^n, \mathbb{R}^n)$. This is much closer to an $\infty$-categorical object since it is a semidirect product of $E_n$ and $O(n)$. If one is taken aback by the presence of the orthogonal group, the place to look for a pure $\infty$-categorical description is the category of framed $n$-manifolds. Ayala-Francis show that excisive functors out of framed $n$-manifold are classified by classical $E_n$-algebras.

Setting aside these philosophical points, let's address the question. The category of $E^{\mathrm{fr}}_n$-algebras is equivalently described as symmetric monoidal functors out of the category $\mathrm{Disk}_n$ of manifolds diffeomorphic to $\bigsqcup_i \mathbb{R}^n$, as $i$ varies. What happens when we drop the symmetric monoidality condition?

In fact, this is a pretty well studied question. In particular, any $n$-manifold $M$ gives rise to a presheaf $E_M^\mathrm{fr}$ which objectwise is given by the embeddings of disjoint unions of disks into $M$. This has the homotopy type of framed configuration spaces, i.e. ordered configuration spaces with a framing of the tangent space at each point in the configuration. These presheaves are the fundamental object of study in embedding calculus (originally known as Goodwillie-Weiss calculus).

In particular, Goodwillie-Klein-Weiss proved that these presheaves contain enough information to recover $\mathrm{Emb}(M,N)$, provided the dimension of $N$ minus the "handle dimension" of $M$ is at least $3$. This latter restriction is obviously a concern if we are trying to recover codimension 0 embeddings, though it is sometimes achieved even in these cases.

The exact nature of the failure of these presheaves to recover $\mathrm{Emb}(M,N)$ in the cases the codimension requirement is not achieved is rather mysterious, but great progress was made by Krannich-Kupers in The Disk Structure Space, where they treated this difference in analogy to the structure spaces that show up in surgery and smoothing theory. If one more completely understood the disk structure spaces, one could hope to use that information to recover the category of smooth $n$-manifolds from the $\infty$-categorical information of the associated presheaf on $\mathrm{Disk}_n$. This is a super interesting, but also incredibly tricky question that I expect we will see progress on in the years to come.