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Finite Semisimple Module 2-Categories

Thibault D. Décoppet

TL;DR

The paper categorifies Ostrik's theorem for multifusion 2-categories by proving that any finite semisimple left module 2-category over a multifusion 2-category C is canonically enriched over C and is equivalent to the 2-category of right modules over a rigid algebra A in C. The enrichment provides a canonical End(M) in C and a Hom-structure giving M the structure of a C-enriched category, with the Hom functor landing in Mod_C(End(M)). When M is a generator, this yields an equivalence M ≃ Mod_C(End(M)), and globally for any finite semisimple M there exists a rigid A with M ≈ Mod_C(A). The results generalize Ostrik’s theorem to the 2-categorical setting and connect finite semisimple module 2-categories with modules over rigid algebras, with implications for separability and dualizability in fusion 2-categories.

Abstract

Let $\mathfrak{C}$ be a multifusion 2-category. We show that every finite semisimple $\mathfrak{C}$-module 2-category is canonically enriched over $\mathfrak{C}$. Using this enrichment, we prove that every finite semisimple $\mathfrak{C}$-module 2-category is equivalent to the 2-category of modules over a rigid algebra in $\mathfrak{C}$.

Finite Semisimple Module 2-Categories

TL;DR

The paper categorifies Ostrik's theorem for multifusion 2-categories by proving that any finite semisimple left module 2-category over a multifusion 2-category C is canonically enriched over C and is equivalent to the 2-category of right modules over a rigid algebra A in C. The enrichment provides a canonical End(M) in C and a Hom-structure giving M the structure of a C-enriched category, with the Hom functor landing in Mod_C(End(M)). When M is a generator, this yields an equivalence M ≃ Mod_C(End(M)), and globally for any finite semisimple M there exists a rigid A with M ≈ Mod_C(A). The results generalize Ostrik’s theorem to the 2-categorical setting and connect finite semisimple module 2-categories with modules over rigid algebras, with implications for separability and dualizability in fusion 2-categories.

Abstract

Let be a multifusion 2-category. We show that every finite semisimple -module 2-category is canonically enriched over . Using this enrichment, we prove that every finite semisimple -module 2-category is equivalent to the 2-category of modules over a rigid algebra in .

Paper Structure

This paper contains 18 sections, 28 theorems, 52 equations, 26 figures.

Table of Contents

  1. Graphical Calculus in Monoidal 2-Categories
  2. Module 2-Categories over Monoidal 2-Categories
  3. Definitions
  4. Properties
  5. Algebras and Modules in Monoidal 2-Categories
  6. Algebras
  7. Modules
  8. Properties
  9. Enrichment of Finite Semisimple Module 2-Categories
  10. Definition
  11. Proof
  12. Comparing Finite Semisimple Module 2-Cate-gories & 2-Categories of Modules
  13. Bimodules associated to Module 2-Functors
  14. Rigid Algebras
  15. Main Theorem
  16. ...and 3 more sections

Key Result

Lemma 1.1.1

Let $F:\mathfrak{A}\rightleftarrows \mathfrak{B}:G$ a coherent 2-adjunction, and $H:\mathfrak{B}\rightarrow \mathfrak{C}$ a 2-functor. For any $B$ in $\mathfrak{B}$, the equality holds in $Hom_{\mathfrak{C}}(H(F(G(B))),H(B))$.

Figures (26)

  • Figure 1: The enriched pentagonator
  • Figure 2: The coherence 2-isomorphism $\mu_{P,Q,R}$
  • Figure 3: The coherence 2-isomorphism $\omega^{\underline{(N,\theta)}}$
  • Figure 4: Axiom b
  • Figure 5: Axiom c
  • ...and 21 more figures

Theorems & Definitions (120)

  • Lemma 1.1.1
  • proof
  • Definition 2.1.1
  • Definition 2.1.3
  • Example 2.1.4
  • Lemma 2.1.5
  • proof
  • Definition 2.1.6
  • Definition 2.1.7
  • Lemma 2.1.8
  • ...and 110 more