Compact Semisimple 2-Categories
Thibault D. Décoppet
TL;DR
This work defines compact semisimple 2-categories over arbitrary fields and proves they are equivalent to $\mathbf{Mod}(\mathcal{C})$ for some finite semisimple tensor 1-category $\mathcal{C}$, via finitely many connected components $\pi_0(\mathfrak{C})$ and local finiteness. It establishes finiteness results over algebraically closed or real closed fields, and analyzes when compactness implies finiteness, tying this to deformation theory through Davydov–Yetter cohomology to show absence of nontrivial deformations for tensor functors and separable tensor 1-categories. A 3-categorical equivalence is proved between finite semisimple tensor data with separable bimodules and compact semisimple 2-categories, together with a Yoneda embedding result and a development of compact semisimple tensor 2-categories, including braided examples and fusion rules. These results set groundwork for higher-dimensional TQFT constructions via dualizability and provide a robust framework for categorified finite semisimple structures over broad fields.
Abstract
Working over an arbitrary field, we define compact semisimple 2-categories, and show that every compact semisimple 2-category is equivalent to the 2-category of separable module 1-categories over a finite semisimple tensor 1-category. Then, we prove that, over an algebraically closed field or a real closed field, compact semisimple 2-categories are finite. Finally, we explain how a number of key results in the theory of finite semisimple 2-categories over an algebraically closed field of characteristic zero can be generalized to compact semisimple 2-categories.