$\newcommand{\Mfld}{\mathsf{Mfld}}
\newcommand{\Space}{\mathsf{Space}}
\newcommand{\Sh}{\operatorname{Sh}}
\newcommand{\PSh}{\operatorname{PSh}}$
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](https://arxiv.org/pdf/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](https://arxiv.org/pdf/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?