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Module categories over graded fusion categories

Ehud Meir, Evgeny Musicantov

TL;DR

This work delivers a comprehensive obstruction-theoretic classification of module categories over a G-graded fusion category C extending a fusion subcategory D by G. By encoding the extension data (c,M,α) and introducing a sequence of obstructions O3, O2, and O3, the authors parameterize indecomposable C-module categories via tuples (N, H, Φ, v, β), together with an isomorphism criterion. They develop Mackey-type results and describe functor categories as equivariantizations, providing criteria for group-theoreticality. An intrinsic description using algebras inside D and C provides practical computational tools and shows how obstructions correspond to splittings of a short exact sequence 1 → Γ → Λ → H → 1. The Tambara–Yamagami examples illustrate the theory in detail, relating module categories, fiber functors, and dual categories, and connecting to Tambara’s results on fiber functors. Overall, the paper advances a unified framework for classifying module categories and their functors over graded extensions of fusion categories, with concrete applications to important examples.

Abstract

Let C be a fusion category which is an extension of a fusion category D by a finite group G. We classify module categories over C in terms of module categories over D and the extension data (c,M,a) of C. We also describe functor categories over C (and in particular the dual categories of C). We use this in order to classify module categories over the Tambara Yamagami fusion categories, and their duals.

Module categories over graded fusion categories

TL;DR

This work delivers a comprehensive obstruction-theoretic classification of module categories over a G-graded fusion category C extending a fusion subcategory D by G. By encoding the extension data (c,M,α) and introducing a sequence of obstructions O3, O2, and O3, the authors parameterize indecomposable C-module categories via tuples (N, H, Φ, v, β), together with an isomorphism criterion. They develop Mackey-type results and describe functor categories as equivariantizations, providing criteria for group-theoreticality. An intrinsic description using algebras inside D and C provides practical computational tools and shows how obstructions correspond to splittings of a short exact sequence 1 → Γ → Λ → H → 1. The Tambara–Yamagami examples illustrate the theory in detail, relating module categories, fiber functors, and dual categories, and connecting to Tambara’s results on fiber functors. Overall, the paper advances a unified framework for classifying module categories and their functors over graded extensions of fusion categories, with concrete applications to important examples.

Abstract

Let C be a fusion category which is an extension of a fusion category D by a finite group G. We classify module categories over C in terms of module categories over D and the extension data (c,M,a) of C. We also describe functor categories over C (and in particular the dual categories of C). We use this in order to classify module categories over the Tambara Yamagami fusion categories, and their duals.

Paper Structure

This paper contains 24 sections, 26 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Decomposition of the module category over the trivial component subcategory. The zeroth obstruction
  4. The first two obstructions
  5. The third obstruction
  6. The isomorphism condition
  7. Functor categories
  8. Mackey's Theorem for module categories
  9. Functor categories
  10. A criterion for an extension to be group theoretical
  11. Functor categories as equivariantizations
  12. An intrinsic description by algebras and modules
  13. The action of $G$ on indecomposable module categories
  14. The outer action of $H$ on the group $Aut_{\mathcal{D}}(\mathcal{N})$. The first two obstructions
  15. The third obstruction
  16. ...and 9 more sections

Key Result

Theorem 1

Let $\mathcal{L}$ be an indecomposable module category over $\mathcal{C}$. There is a subgroup $H<G$, and an indecomposable $\mathcal{C}_{H}=\bigoplus_{a\in H}\mathcal{C}_{a}$ module category $\mathcal{N}$ which remains indecomposable over $\mathcal{D}$ such that $\mathcal{L}\cong Ind_{\mathcal{C}_{

Theorems & Definitions (45)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark
  • Theorem 4
  • Lemma 5
  • proof
  • Remark 6
  • Corollary 7
  • Lemma 8
  • ...and 35 more