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A lifting theorem for Grothendieck-Verdier categories

Max Demirdilek

TL;DR

This work identifies a lifting principle for Grothendieck-Verdier (GV) categories: a conservative lax monoidal functor $U: \mathcal{C} \to \mathcal{D}$ into a GV-category, equipped with a Frobenius form, lifts the GV-duality from $\mathcal{D}$ to $\mathcal{C}$, making $U$ a GV-functor. It extends the landscape by establishing a 2-equivalence between GV-categories and linearly distributive categories with negation (LDN), and a braided version of this equivalence, thereby providing a categorical bridge to Frobenius linearly distributive functors. The lifting theorem is then applied to Hopf monads and Hopf algebroids, as well as to categories of (bi)modules and local modules, yielding GV-structures in a range of settings including smash product algebras, skew group algebras, and enveloping algebras of Lie–Rinehart algebras. These results unify and extend prior work on duality in GV-categories and offer systematic methods for constructing GV-structures in module-theoretic contexts with concrete algebraic examples.

Abstract

We identify additional structure on a conservative lax monoidal functor from a closed monoidal category $\mathcal{C}$ to a Grothendieck-Verdier category $\mathcal{D}$, such that the Grothendieck-Verdier structure of $\mathcal{D}$ lifts to $\mathcal{C}$ and the functor becomes Frobenius linearly distributive. As an application, we recover and extend conditions under which modules over Hopf monads and Hopf algebroids inherit Grothendieck-Verdier structures. We also characterize when categories of bimodules, modules, and local modules over (commutative) algebras internal to a Grothendieck-Verdier category admit such structures. Our results apply to quantales, smash product algebras, skew group algebras, and enveloping algebras of Lie-Rinehart algebras. For applications of the lifting theorem, we construct a strict $2$-equivalence between a $2$-category of Grothendieck-Verdier categories and one of linearly distributive categories with negation, and extend this $2$-equivalence to the braided setting.

A lifting theorem for Grothendieck-Verdier categories

TL;DR

This work identifies a lifting principle for Grothendieck-Verdier (GV) categories: a conservative lax monoidal functor into a GV-category, equipped with a Frobenius form, lifts the GV-duality from to , making a GV-functor. It extends the landscape by establishing a 2-equivalence between GV-categories and linearly distributive categories with negation (LDN), and a braided version of this equivalence, thereby providing a categorical bridge to Frobenius linearly distributive functors. The lifting theorem is then applied to Hopf monads and Hopf algebroids, as well as to categories of (bi)modules and local modules, yielding GV-structures in a range of settings including smash product algebras, skew group algebras, and enveloping algebras of Lie–Rinehart algebras. These results unify and extend prior work on duality in GV-categories and offer systematic methods for constructing GV-structures in module-theoretic contexts with concrete algebraic examples.

Abstract

We identify additional structure on a conservative lax monoidal functor from a closed monoidal category to a Grothendieck-Verdier category , such that the Grothendieck-Verdier structure of lifts to and the functor becomes Frobenius linearly distributive. As an application, we recover and extend conditions under which modules over Hopf monads and Hopf algebroids inherit Grothendieck-Verdier structures. We also characterize when categories of bimodules, modules, and local modules over (commutative) algebras internal to a Grothendieck-Verdier category admit such structures. Our results apply to quantales, smash product algebras, skew group algebras, and enveloping algebras of Lie-Rinehart algebras. For applications of the lifting theorem, we construct a strict -equivalence between a -category of Grothendieck-Verdier categories and one of linearly distributive categories with negation, and extend this -equivalence to the braided setting.
Paper Structure (37 sections, 70 theorems, 156 equations, 1 figure)

This paper contains 37 sections, 70 theorems, 156 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Preliminaries
  4. Grothendieck-Verdier functors
  5. Closed monoidal categories
  6. Closed monoidal functors
  7. Grothendieck-Verdier categories as closed monoidal categories
  8. Grothendieck-Verdier functors
  9. A $2$-equivalence involving linearly distributive categories
  10. Linearly distributive categories
  11. A $2$-equivalence between $\mathsf{GV}$ and $\mathsf{LDN}$
  12. From $\mathsf{GV}$ to $\mathsf{LDN}$
  13. From $\mathsf{LDN}$ to $\mathsf{GV}$
  14. Pivotality, braidings, and another $2$-equivalence
  15. Pivotality
  16. ...and 22 more sections

Key Result

Lemma 2.5

(Cf. EilenbergKellyFun). The evaluation $\operatorname{ev}^X$ and coevaluation $\operatorname{coev}^X$ are both extranatural in the component $X\in \mathcal{C}$. This means that for all $X,Y,Z \in \mathcal{C}$ and every $f\in \operatorname{Hom}_{\mathcal{C}}(X,Y)$, Analogous identities hold for any right closed monoidal category.

Figures (1)

  • Figure :

Theorems & Definitions (222)

  • Definition 2.1
  • Remark 2.2: Terminology
  • Remark 2.3: Coclosedness
  • Remark 2.4: Notation
  • Lemma 2.5
  • Remark 2.6: Internal composition
  • Lemma 2.7
  • Remark 2.8: Internal algebra
  • Lemma 2.9
  • Lemma 2.10
  • ...and 212 more