Lurie (On the Classification of Topological Field Theories), with some corrections by Calaque and Scheimbauer (A note on the $(\infty,n)$-category of cobordisms), famously constructed a symmetric monoidal $(\infty,n)$-category $\mathrm{Bord}_n$ of $n$-dimensional smooth (co)bordisms.

Actually, the construction produces the following. Saying "such and such thing is an $(\infty,n)$-category" requires choosing a *model* of the $\infty$-category $\{\text{$(\infty,n)$-categories}\}$. By a "model" I mean a model category; "$(\infty,n)$-category" is then defined to mean fibrant-cofibrant object in the model. To foreshadow the question I'm going to get to, let me emphasize that a random object in the model category *is not* an $(\infty,n)$-category according to the model. Rather, it determines an $(\infty,n)$-category by (co)fibrant replacement.

Lurie and Calaque–Scheimbauer use the model of $(\infty,n)$-categories called "complete $n$-fold Segal spaces". The objects in the model are $n$-fold simplicial spaces, and the words "Segal" and "complete" refer to specific fibrancy conditions. Speaking approximately, "Segal" is a version of the requirement that composition be well-defined and single-valued: it rules out, for example, a situation where you have morphisms $f : X\to Y$ and $g : Y \to Z$, but no composition "$g\circ f$". "Completeness" has to do with invertibility: it says that a weak form of invertibility implies a strong form of invertibility. Fibrant replacement for the Segal condition goes through and adds in formal compositions of previously-incomposable pairs. Fibrant replacement for the completeness condition goes through and adds formal strong inverses to weakly-invertible things.

With this choice in mind, Lurie and Calaque–Scheimbauer build an $n$-fold simplicial space which deserves the name $\mathrm{Bord}_n$ in the following sense: its entries are moduli spaces of bordisms. They prove that $\mathrm{Bord}_n$ satisfies the Segal condition. However, they also emphasize that, for large enough $n$, **the $n$-fold simplicial space $\mathrm{Bord}_n$ of Lurie and Calaque–Scheimbauer is not complete.** The issue has to do with the h-cobordism theorem, which can be used to give morphisms that I am calling "weakly but not strongly invertible".

For the purposes of stating the Cobordism Hypothesis, this in some sense "doesn't matter". First, as I foreshadowed above, $\mathrm{Bord}_n$ has a completion $\widehat{\mathrm{Bord}}_n$: when people talk about "the $(\infty,n)$-category of cobordisms" and cite Lurie or Calaque–Scheimbauer, they really mean $\widehat{\mathrm{Bord}}_n$ and not $\mathrm{Bord}_n$. Second, if $\mathcal{C}$ is any true $(\infty,n)$-category, i.e. a complete $n$-fold Segal space, then $\hom(\mathrm{Bord}_n,\mathcal{C}) = \hom(\widehat{\mathrm{Bord}}_n,\mathcal{C})$. So a universal-property characterization like "$\hom(\mathrm{Bord}_n,\mathcal{C}) = \{\text{fully-dualizable objects in $\mathcal C$}\}$" will hold for both if it holds for either.

On the other hand, the replacement procedure to complete $\mathrm{Bord}_n \leadsto \widehat{\mathrm{Bord}}_n$ could be quite aggressive: in general, completion might require adding in lots of extra morphisms. For instance, one thing that I know that completion does is that it takes any h-cobordism $W$ between $M$ and $N$, slices it up $W$ parallel to the $M$- and $N$-boundaries, and then reads the slices as a new "smooth" family of manifolds connecting $M$ and $N$. In other words, this procedure goes through and potentially-aggressively modifies the moduli spaces of manifolds from what you originally thought they were.

The general version of my question is:

Just how aggressive is the completion $\mathrm{Bord}_n \leadsto \widehat{\mathrm{Bord}}_n$? How much does it add to the moduli spaces of $k$-dimensional bordisms?

Here is a specific version of my question. The existence of h-cobordisms is related to the existence of *manifold factors*, which are non-manifold spaces $X$, like the dogbone space, such that $X \times \mathbb R \cong \mathbb R^N$, or more generally $X \times M \cong N$ for some manifolds $M$ and $N$.

Do manifold factors show up as $k$-morphisms in $\widehat{\mathrm{Bord}}_n$? Do fully-extended TFTs assign values to manifold factors?

flagged∞-categories, that is pairs $(C,X→C)$ where $C$ is an ∞-category and $X$ is an ∞-groupoid with an essentially surjective map to $C$. Then "completion" corresponds to forgetting about $X$. Under this equivalence, the Segal space of bordisms corresponds to the flagged ∞-category $M→\mathrm{Bord}$ where $\mathrm{Bord}$ is the bordism category and $M$ is the ∞-groupoid of manifolds and diffeomorphisms. I expect the situation to be the same for higher bordism categories. $\endgroup$ – Denis Nardin Dec 3 '20 at 20:23