Topological defects and higher-categorical structures

Abstract

I will discuss some (higher-)categorical structures present in three-dimensional topological field theories that include topological defects of any codimension. The emphasis will be on two topics: (1) For Reshetikhin-Turaev type theories, regarded as 3-2-1-extended TFTs, I will explain why codimension-1 boundaries and defects form bicategories of module categories over suitable fusion categories. In the case of defects separating three-dimensional regions supporting the same theory, the relevant fusion category $A$ is the modular tensor category underlying that theory, while for defects separating two theories of Turaec-Viro type with underlying fusion categories $A_1$ and $A_2$, respectively, $A$ is the the Deligne product $A_1 \boxtimes A_2^{op}$. (2) I will indicate the building blocks of a generalization of the TV-BW state-sum construction to theories with defects. Making use of ends and coends, various aspects of this construction can be formulated without requiring semisimplicity.