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Surface Diagrams for Frobenius Algebras and Frobenius-Schur Indicators in Grothendieck-Verdier Categories

Max Demirdilek, Christoph Schweigert

TL;DR

The paper develops a three-dimensional surface-diagram calculus for Grothendieck-Verdier categories to manage non-invertible coherence data and extends Frobenius theory to linearly distributive (LD) contexts. It defines LD-Frobenius algebras, proves core Frobenius relations (including their non-independence) and establishes an equivalence between module and comodule categories in this LD setting. It then introduces higher Frobenius-Schur indicators in pivotal GV-categories and proves their invariance under pivotal Frobenius-LD-equivalences, offering a robust, graphical toolkit for handling GV- and LD-structures. The work provides STL/HOM resources for 3D models and graphical proofs, with concrete relevance to bimodule categories, Hopf algebroids, and VOA module categories, and suggests applications to logarithmic CFT and related representation theory.

Abstract

Grothendieck-Verdier categories (also known as $\ast$-autonomous categories) generalize rigid monoidal categories, with notable representation-theoretic examples including categories of bimodules, modules over Hopf algebroids, and modules over vertex operator algebras. In this paper, we develop a surface-diagrammatic calculus for Grothendieck-Verdier categories, extending the string-diagrammatic calculus of Joyal and Street for rigid monoidal categories into a third dimension. This extension naturally arises from the non-invertibility of coherence data in Grothendieck-Verdier categories. We show that key properties of Frobenius algebras in rigid monoidal categories carry over to the Grothendieck-Verdier setting. Moreover, we introduce higher Frobenius-Schur indicators for suitably finite $k$-linear pivotal Grothendieck-Verdier categories and prove their invariance under pivotal Frobenius linearly distributive equivalences. The proofs are carried out using the surface-diagrammatic calculus. To facilitate the verification of some of our results, we provide auxiliary files for the graphical proof assistant homotopy$.$io.

Surface Diagrams for Frobenius Algebras and Frobenius-Schur Indicators in Grothendieck-Verdier Categories

TL;DR

The paper develops a three-dimensional surface-diagram calculus for Grothendieck-Verdier categories to manage non-invertible coherence data and extends Frobenius theory to linearly distributive (LD) contexts. It defines LD-Frobenius algebras, proves core Frobenius relations (including their non-independence) and establishes an equivalence between module and comodule categories in this LD setting. It then introduces higher Frobenius-Schur indicators in pivotal GV-categories and proves their invariance under pivotal Frobenius-LD-equivalences, offering a robust, graphical toolkit for handling GV- and LD-structures. The work provides STL/HOM resources for 3D models and graphical proofs, with concrete relevance to bimodule categories, Hopf algebroids, and VOA module categories, and suggests applications to logarithmic CFT and related representation theory.

Abstract

Grothendieck-Verdier categories (also known as -autonomous categories) generalize rigid monoidal categories, with notable representation-theoretic examples including categories of bimodules, modules over Hopf algebroids, and modules over vertex operator algebras. In this paper, we develop a surface-diagrammatic calculus for Grothendieck-Verdier categories, extending the string-diagrammatic calculus of Joyal and Street for rigid monoidal categories into a third dimension. This extension naturally arises from the non-invertibility of coherence data in Grothendieck-Verdier categories. We show that key properties of Frobenius algebras in rigid monoidal categories carry over to the Grothendieck-Verdier setting. Moreover, we introduce higher Frobenius-Schur indicators for suitably finite -linear pivotal Grothendieck-Verdier categories and prove their invariance under pivotal Frobenius linearly distributive equivalences. The proofs are carried out using the surface-diagrammatic calculus. To facilitate the verification of some of our results, we provide auxiliary files for the graphical proof assistant homotopyio.

Paper Structure

This paper contains 33 sections, 35 theorems, 69 equations, 57 figures.

Table of Contents

  1. Introduction
  2. Categorical preliminaries
  3. Surface diagrams for monoidal bicategories
  4. The macrocosm $\mathcal{V}{\textnormal{-}}\mathsf{Cat}$
  5. Categories
  6. Functors
  7. The canvas.
  8. Graphical depiction.
  9. Compositions.
  10. Natural transformations.
  11. The canvas.
  12. Graphical depiction.
  13. Examples.
  14. Compositions.
  15. Conventions and terminology.
  16. ...and 18 more sections

Key Result

Proposition 2.16

Let $\mathcal{C}$ and $\mathcal{D}$ be monoidal categories. Any strong monoidal functor ${F\colon\mathcal{C}\rightarrow \mathcal{D}}$ is a Frobenius monoidal functor.

Figures (57)

  • Figure 1: Depicting categories.
  • Figure 2: Depicting functors.
  • Figure 3: Cartesian product $\times$ of functors.
  • Figure 4: Composition $\ast_0$ of functors.
  • Figure 5: The $3$-framed canvas.
  • ...and 52 more figures

Theorems & Definitions (142)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 132 more