(Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories

Abstract

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong $(\infty, n)$-functors. We construct a double $(\infty,n)$-category built out of the target $(\infty, n)$-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of $E_d$-algebras in a symmetric monoidal $(\infty, n)$-category $\mathcal{C}$ to an $(\infty, n+d)$-category using the higher morphisms in $\mathcal{C}$.