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.