The n-Category Café

The first lecture considered an example of a $2$-dimensional TFT constructed from a finite group $\Gamma$ by assigning to the point the category of modules of the group algebra $\mathbb{C}[\Gamma]$. The second lecture covered categorical versions of the group algebra for a complex reductive group $G$. This was in preparation for a discussion of topological field theories associated to $G$. These require higher categorical constructions, namely, $2$-categories of $G$-module categories assigned to the point.

This lecture focuses on two versions of $G$-module categories: algebraic $G$-categories and smooth $G$-categories. By a result of Ben-Zvi, Francis and Nadler, assigning the $2$-category of algebraic $G$-categories to the point defines a $2$-dimensional TFT. Assigning the $2$-category of smooth $G$-categories to the point only defines a $1$-dimensional TFT. A modified version extends up to $2$-manifolds. This modification is defined by assigning to the point the $2$-category of $\mathcal{H}$-mod where $\mathcal{H}$ is the finite Hecke category. From a physics perspective all of these are part of a $3$-dimensional gauge theory.

Continue reading about lecture 3 here.