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Rigid and Separable Algebras in Fusion 2-Categories

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arxiv 2205.06453 v4 pith:TZESH234 submitted 2022-05-13 math.QA math.CT

Thibault D. Décoppet This is my paper · ORCID
classification math.QAmath.CT
keywords rigidcategoriesalgebraalgebrasfusionseparablecategorybimodules
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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.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Classification of symmetric fusion categories over $\mathbb{R}$

    math.QA 2026-08 conditional novelty 7.0 of 10

    Every symmetric fusion category over R is equivalent to the semi-linear super representation category of a Z2-graded finite super group.

  2. The Classification of 3+1d Symmetry Enriched Topological Order

    math-ph 2025-09 conditional novelty 7.0 of 10

    Finite-group-enriched 3+1d topological orders are classified by 2SVect-enriched G-crossed braided fusion 2-categories, with gauging obstructions living in SW^5(BG).

  3. Compact Semisimple Tensor 2-Categories are Morita Connected

    math.QA 2024-12 conditional novelty 7.0 of 10

    Every compact semisimple tensor 2-category is Morita equivalent to a connected one over arbitrary fields of characteristic zero, with applications to Witt groups and Galois cohomology.

  4. On \'Etale Algebras and Bosonic Fusion 2-Categories

    math.CT 2024-11 conditional novelty 7.0 of 10

    Connected and Lagrangian étale algebras in Z_1(2Vect^π_G) are classified by subgroups, braided fusion categories with group actions, and 4-group morphism data; this yields a parametrization of bosonic fusion 2-categories.

  5. Frobenius Algebras and Dual Bimodules in Monoidal 2-Categories

    math.QA 2026-06 unverdicted novelty 6.5 of 10

    Explicit construction of dual bimodules from Frobenius algebras in monoidal 2-categories, with promotion of coherent duals and proof that special Frobenius algebras in 2Vect are rigid.

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