Rigid and Separable Algebras in Fusion 2-Categories

TL;DR

This work generalizes rigidity from fusion 1-categories to rigid algebras in finite semisimple monoidal 2-categories, and analyzes the induced 2-categories of modules and bimodules. It establishes precise criteria for rigidity and separability: rigidity of $A$ is equivalent to the existence of right adjoints in $\mathbf{Bimod}_{\mathfrak{C}}(A)$, while separability is equivalent to finite semisimplicity of the center $Z(A)$ and to nonvanishing of the dimension $\mathrm{Dim}_{\mathfrak{C}}(A)$ for connected algebras over suitable fields. The dimension invariant $\mathrm{Dim}_{\mathfrak{C}}(A)$ provides a practical test for separability, recovering the classical categorical dimension when $\mathfrak{C}=\mathbf{2Vect}$ and connecting to dualizability questions for fusion 2-categories. The results apply to central examples such as $\mathbf{2Vect}_G$, $\mathbf{Mod}(\mathcal{B})$, and $\mathscr{Z}(\mathbf{2Vect}_G)$, and they clarify when module/bimodule 2-categories are compact semisimple, informing higher-dimensional dualizability and topological field theory constructions.

Abstract

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include $G$-graded fusion 1-categories, and $G$-crossed fusion 1-categories. We explore the properties of the 2-categories of modules and of bimodules over a rigid algebra, by giving a criterion for the existence of right and left adjoints. Then, we consider separable algebras, which are particularly well-behaved rigid algebras. Specifically, given a fusion 2-category, we prove that the 2-categories of modules and of bimodules over a separable algebra are finite semisimple. Finally, we define the dimension of a connected rigid algebra in a fusion 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.

Figures (59)

• Figure 1: Equality (Part 1)
• Figure 2: Equality (Part 2)
• Figure 3: Equality (Part 3)
• Figure 4: Equality (Part 4)
• Figure 5: Equality (Part 5)

Theorems & Definitions (87)

• Definition 1.2.1
• Lemma 1.2.2
• proof
• Definition 1.2.3
• Lemma 1.2.4
• proof
• Remark 1.2.5
• Definition 1.2.6
• Definition 1.2.7
• Lemma 1.2.8