Duals and adjoints in higher Morita categories

Abstract

We study duals for objects and adjoints for $k$-morphisms in $\operatorname{Alg}_n(\mathcal{S})$, an $(\infty,n+N)$-category that models a higher Morita category for $E_n$ algebra objects in a symmetric monoidal $(\infty,N)$-category $\mathcal{S}$. Our model of $\operatorname{Alg}(\mathcal{S})$ uses the geometrically convenient framework of factorization algebras. The main result is that $\operatorname{Alg}_n(\mathcal{S})$ is fully $n$-dualizable, verifying a conjecture of Lurie. Moreover, we unpack the consequences for a natural class of fully extended topological field theories and explore $(n+1)$-dualizability.