Weak Fusion 2-Categories
Thibault D. Décoppet
TL;DR
The paper introduces a weakening of fusion 2-categories and develops a robust framework for multifusion 2-categories within finite semisimple, k-linear monoidal 2-categories. It establishes equivalence with the Douglas–Reutter strict notion, analyzes adjoints and coherent duals, and proves that connected fusion 2-categories correspond precisely to braided fusion categories, with module-2-category constructions coding the braided structure. It also provides explicit fusion rules for connected fusion 2-categories arising from pointed braided fusion categories via relative Deligne tensor products, illustrating the dependence on the chosen braiding. The results bridge higher-category theory and braided tensor categories, yielding concrete tools for computing fusion data and informing potential applications in higher representation theory and topological quantum field theories.
Abstract
We introduce a weakening of the notion of fusion 2-category given in arXiv:1812.11933. Then, we establish a number of properties of (multi)fusion 2-categories. Finally, we describe the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories.