Duals and adjoints in higher Morita categories

Abstract

We study duals for objects and adjoints for $k$-morphisms in ${\mathrm{Alg}}_n(\mathcal{S})$, 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 $\mathrm{Alg}(\mathcal{S})$ uses the geometrically convenient framework of factorization algebras. The main result is that ${\mathrm{Alg}}_n(\mathcal{S})$ 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.

44 pages, many pictures. Minor changes, now printable in grayscale. Comments welcome!