Colloque 2015 du GDR 2875, Topologie Algébrique et Applications

(Op)lax natural transformations for higher categories, relative quantum field theories, and the "even higher" Morita category

23 oct. 2015, 11:40

50m

Amphi Schwartz, bat. 1R3 (Institut de Mathématiques de Toulouse)

Exposé de recherche sur invitation TopAlg

Orateur

Dr Claudia Scheimbauer (Max Planck Institute for Mathematics, Bonn)

Description

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, known as transfors, between strong $(\mathrm{\infty },n)$-functors. It is given by a double $(\mathrm{\infty },n)$-category built out of the target $(\mathrm{\infty },n)$-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 $(\mathrm{\infty },n)$-category $\mathcal{C}$ to an even higher level using the higher morphisms of $\mathcal{C}$.