Let $\mathcal{B}$ be a braided ($n-1$)-category. I will assume that $\mathcal{B}$ is a fully-dualizable object in some $n+1$-category of braided ($n-1$)-categories. Hence, from $\mathcal{B}$, using the cobordism hypothesis, one gets a TQFT $\int\_{\Box}\mathcal{B}$.


I have read recently that $$\int\_{S^{n-1}\_b}\mathcal{B} = \Omega^{n-1}\mathcal{B},$$
where $S^{n-1}\_b$ denotes the $(n-1)$-sphere with its blackboard framing, and $\Omega^{n-1}\mathcal{B}$ denotes the $1$-category of $(n-2)$- and $(n-1)$-endomorphisms of the monoidal unit of $\mathcal{B}$.


Why is that true?



Factorization homology of a braided (n-1)-category on an (n-1)-sphere
I think this is an error in my paper. Thank you for finding it. The overall result is correct, but the proof is wrong as written. To correct it, I need to replace "TQFT" with "relative TQFT", and replace $S^{n-1}\_b$ with the pair $(D^n, S^{n-1}\_b)$, and the rest is correct. In detail:
$\newcommand\cB{\mathcal{B}}\newcommand\cX{\mathcal{X}}$


I want to appeal to a "state-operator correspondence", which for an absolute (aka nonanomalous) $(n+1)$-dimensional TQFT $\cX$ says that the $m$-category of operators of dimension $\leq m$ is $\cX(S\_b^{n-m})$. Note that, counting dimensions, this is indeed an $m$-category.


But I wanted to apply this to the relative (aka anomalous) $(n+1)$-dimensional TQFT $\cB$, and its compactification $\cB^2 = \int\_{S^1\_b}\cB$. This $(n+1)$-dimensional TQFT is relative to an $(n+2)$-dimensional TQFT which depends only on the Morita equivalence class of $\cB$, and my notation did not distinguish these.


So I need to tell you about the state-operator correspondence for relative TQFTs. To say it, let me remind that a relative TQFT is the type of thing that can eat a "cobordism with boundary". This is a manifold with two types of boundaries: you can stitch cobordisms together along one of the types of boundaries, and the other type of boundary is marked by a boundary condition.


I don't have good notation. I will write $\cB\_\partial$ for the relative TQFT in question. Suppose $M^m$ is an $m$-dimensional cobordism and I decide that $N^{m-1} \subset \partial M$ is where I will place the boundary condition (and I leave the rest of $\partial M$ as stitchable); then I will write $\cB\_\partial(M, N)$, and I won't try to use footnotes. Any absolute $(n+1)$-dimensional TQFT $\cX$ gives a relative $(n+1)$-dimensional TQFT $\cX\_{\partial}$ in which $\cX\_\partial(M^m,N^{m-1}) = \cX(N^{m-1})$ depends only on $N$. On the other hand, any relative TQFT gives an absolute TQFT in which you only use cobordisms without boundary.


With this all said, the state-operator correspondence for relative $(n+1)$-dimensional TQFTs asserts that the boundary operators of dimension $\leq m$ in $\cX\_\partial$ is $\cX\_\partial(D^{n+1-m}, S^{n-m}\_b)$, where $D^{n+1-m}$ is the disk and $S^{n-m}$ is its full boundary. So this is a cobordism from $\emptyset$ to $\emptyset$.


Ok, now I can compute. I want to show that the multifusion $(n-1)$-category $\cB^e$ is fusion, which is to say that the relative $n$-dimensional TQFT $\cB^e\_\partial$ satisfies $\cB^e\_\partial(D^n , S^{n-1}) = \mathbb{C}$. As in my paper, use that $\cB^e\_\partial$ is a compactification on $S^1$ on $\cB\_\partial$, so that what we want to compute is $\cB\_\partial(D^n \times S^1, S^{n-1} \times S^1)$. But I can do this by first compactifying on $(D^n, S^{n-1})$ to get an absolute 2D TQFT defined by the 1-category $\cB\_\partial(D^n, S^{n-1})$, and we already said that that is precisely the 1-category of boundary lines, i.e. what I called $\Omega^{n-1}\cB$.


