(Op)lax natural transformations for higher categories and two applications

A relative (also called twisted) quantum field theory should be some transformation between quantum field theories, which themselves are symmetric monoidal functors out of a space-time category. In examples, the notion of natural transformation turns out to be too strong, making it necessary to relax it. In joint work with Theo Johson-Freyd we provide a framework for both lax and oplax transformations and their higher analogs called transfors between strong $(\infty, n)$-functors. It is given by a double $(\infty,n)$-category built out of the target $(\infty, n)$-category that we call its (op)lax square, which governs the desired diagrammatics. Lax or oplax transfors then are functors into parts of the oplax square. Finally, I will explain how to use the (op)lax square to extend the construction of the higher Morita category of $E_d$-algebras in an $(\infty,n)$-category $\mathcal C$ to an even higher level using the higher morphisms of $\mathcal C$.