Cobordism hypothesis

Fix an integer $n\geq 1$.

Let $\mathrm{Bord}_n^{\mathrm{fr}}$ denote the symmetric monoidal $(\infty,n)$-category of framed bordisms: its objects are $0$-dimensional framed manifolds, for $1\leq k\leq n$ its $k$-morphisms are $k$-dimensional framed bordisms between $(k-1)$-dimensional framed bordisms, and its $(n+j)$-morphisms for $j\geq 1$ are diffeomorphisms and higher isotopies of $n$-dimensional framed bordisms; the symmetric monoidal structure is given by disjoint union, with unit the empty manifold. Write $\mathrm{pt}_+ \in \mathrm{Bord}_n^{\mathrm{fr}}$ for the point equipped with its positive framing.

Let $\mathcal{C}$ be a symmetric monoidal $(\infty,n)$-category satisfying:

1. every object of $\mathcal{C}$ is fully dualizable, and 2. for every $k$ with $1\leq k\leq n-1$, every $k$-morphism of $\mathcal{C}$ admits both a left adjoint and a right adjoint.

Let $\mathrm{Fun}^{\otimes}(\mathrm{Bord}_n^{\mathrm{fr}},\mathcal{C})$ denote the $(\infty,n)$-category of symmetric monoidal functors from $\mathrm{Bord}_n^{\mathrm{fr}}$ to $\mathcal{C}$, and let $\mathcal{C}^{\sim}$ denote the underlying $\infty$-groupoid of $\mathcal{C}$, obtained by discarding all non-invertible $k$-morphisms for $1 \le k \le n$.

Then $\mathrm{Fun}^{\otimes}(\mathrm{Bord}_n^{\mathrm{fr}},\mathcal{C})$ is an $\infty$-groupoid, and the evaluation map

$$\mathrm{Fun}^{\otimes}(\mathrm{Bord}_n^{\mathrm{fr}},\mathcal{C}) \longrightarrow \mathcal{C}^{\sim}, \qquad Z \longmapsto Z(\mathrm{pt}_+),$$

is an equivalence of $\infty$-groupoids. In particular, $Z \mapsto Z(\mathrm{pt}_+)$ induces a bijection between the set of equivalence classes of symmetric monoidal functors $\mathrm{Bord}_n^{\mathrm{fr}} \to \mathcal{C}$ and the set of equivalence classes of objects of $\mathcal{C}$.