act on the classifying space for the underlying $n$-groupoid of the fully dualizable objects in ${\mathcal{C}}$). When $n=1$, this action is simply given by the involution that takes a dualizable object $C \in {\mathcal{C}}$ to its dual $C^{\vee}$.

We can use the action of the orthogonal group to formulate an analog of Theorem \[bd\] for more general types of manifolds:

\[bdo\] Let ${\mathcal{C}}$ be a symmetric monoidal $n$-category. The construction $F \mapsto F(\ast)$ establishes a bijection between the set of isomorphism classes of symmetric monoidal functors ${\mathbf{Bord}}_{n}^{\operatorname{SO}} \rightarrow {\mathcal{C}}$ with the set of isomorphism classes of (homotopy) fixed points for the action of the group $\operatorname{SO}(n)$ on the fully dualizable objects of ${\mathcal{C}}$.

Theorem \[bdo\] has an obvious analog for other types of manifolds: unoriented manifolds, spin manifolds, and so forth.

\[alg\] Let $n=2$, and let ${\mathcal{C}}$ denote the $2$-category ${\mathcal{A}lg}$ of (complex) algebras, bimodules, and intertwiners. Every object of ${\mathcal{C}}$ admits a dual: the dual of an algebra $A$ is the opposite algebra $A^{op}$, where both the evaluation and coevaluation maps are given by $A$ (regarded as a $({\mathbb
C}, A \otimes A^{op})$-bimodule. An algebra $A$ is fully dualizable if and only if $A$ is dualizable both as a 