On dualizability and invertibility in the higher Morita category

We prove a conjecture of Lurie characterizing $(n+1)$-dualizability in higher Morita categories of ${\mathbb{E}}_n$-algebras in terms of dualizability over certain factorization homologies. A key ingredient is a higher Morita category based on the recently developed framework of pointless factorization algebras of Karlsson and the first author. We also verify an invertibility conjecture of Brochier–Jordan–Safranov–Snyder as an immediate corollary of our main result. Moreover, we prove a relative version of the dualizability conjecture, yielding a new criterion for relative/twisted field theories. We give some examples, including Dirichlet and Neumann relative theories.