On dualizability of braided tensor categories

TL;DR

This paper investigates the dualizability properties of various higher categories of tensor and braided tensor categories, establishing their dualizability levels and implications for topological field theories.

Contribution

It proves dualizability results for categories of tensor and braided tensor categories, enabling the construction of associated topological field theories in different dimensions.

Findings

The 3-category of rigid tensor categories with enough compact projectives is 2-dualizable.

The 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable.

The 4-category of braided fusion categories in characteristic zero is 4-dualizable.

Abstract

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produce a framed 3-dimensional local TFT attached to the category of representations of a quantum group at any value of $q$.