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Equivariant Eilenberg-Watts theorem for module coalgebras

Taiki Shibata, Kenichi Shimizu

TL;DR

This work extends the classical Takeuchi Eilenberg-Watts correspondence to an equivariant setting for module coalgebras over a bialgebra $H$, proving the fundamental equivalence ${}^C_H\mathfrak{M}^D \simeq \mathscr{L}_{\mathfrak{M}^H}^{\mathrm{lax}}(\mathfrak{M}^C,\mathfrak{M}^D)$ with source objects being $C$-$D$-bicomodules in ${}_H\mathfrak{M}$. The authors develop explicit constructions tying $H$-actions on bicomodules to lax $\mathfrak{M}^H$-module functor structures, and they provide a suite of variants and applications. They derive sharp consequences for Hopf and Yetter-Drinfeld modules, including a condition under which lax equals strong and the fundamental Hopf-module theorem, along with a Yetter-Drinfeld equivalence in the Hopf setting. The paper also presents a bicategorical perspective, establishing 2-equivalences between module categories over ${\mathfrak{M}}^H$ and ${}_H\mathfrak{M}$ and formulating an equivariant Morita-Takeuchi theory, further connecting subcoalgebra data to module-subcategory structure. Overall, the results illuminate a robust duality between equivariant coalgebra filtrations and lax module functorial behavior, with structural insights for quantum group representations and categorical Morita theory.

Abstract

For coalgebras $C$ and $D$, Takeuchi proved that the category of linear functors from $\mathfrak{M}^C$ to $\mathfrak{M}^D$ preserving small coproducts is equivalent to the category of $C$-$D$-bicomodules, where $\mathfrak{M}^C$ for a coalgebra $C$ means the category of right $C$-comodules. We formulate and prove an equivariant version of this result for module coalgebras over a bialgebra. As an application, for a bialgebra $H$, we establish an equivalence of the 2-category of a particular class of module categories over the monoidal category $\mathfrak{M}^H$ and the 2-category of a particular class of module categories over the monoidal category ${}_H\mathfrak{M}$ of left $H$-modules.

Equivariant Eilenberg-Watts theorem for module coalgebras

TL;DR

This work extends the classical Takeuchi Eilenberg-Watts correspondence to an equivariant setting for module coalgebras over a bialgebra , proving the fundamental equivalence with source objects being --bicomodules in . The authors develop explicit constructions tying -actions on bicomodules to lax -module functor structures, and they provide a suite of variants and applications. They derive sharp consequences for Hopf and Yetter-Drinfeld modules, including a condition under which lax equals strong and the fundamental Hopf-module theorem, along with a Yetter-Drinfeld equivalence in the Hopf setting. The paper also presents a bicategorical perspective, establishing 2-equivalences between module categories over and and formulating an equivariant Morita-Takeuchi theory, further connecting subcoalgebra data to module-subcategory structure. Overall, the results illuminate a robust duality between equivariant coalgebra filtrations and lax module functorial behavior, with structural insights for quantum group representations and categorical Morita theory.

Abstract

For coalgebras and , Takeuchi proved that the category of linear functors from to preserving small coproducts is equivalent to the category of --bicomodules, where for a coalgebra means the category of right -comodules. We formulate and prove an equivariant version of this result for module coalgebras over a bialgebra. As an application, for a bialgebra , we establish an equivalence of the 2-category of a particular class of module categories over the monoidal category and the 2-category of a particular class of module categories over the monoidal category of left -modules.

Paper Structure

This paper contains 21 sections, 21 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Modules over a monoidal category
  5. Canonical $\mathbf{Vec}$-action
  6. Canonical $\mathbf{Vec}$-module structure
  7. Takeuchi's equivalence
  8. Eilenberg-Watts equivalence
  9. Equivariant Eilenberg-Watts theorem
  10. Main result
  11. Proof of Theorem \ref{['thm:equivariant-EW']}
  12. Variants of Theorem \ref{['thm:equivariant-EW']}
  13. Applications to Hopf and Yetter-Drinfeld modules
  14. When is a lax module functor strong?
  15. The fundamental theorem for Hopf modules
  16. ...and 6 more sections

Key Result

Lemma 2.3

Let $\mathcal{M}$ and $\mathcal{N}$ be linear categories admitting small coproducts. Then the category of linear functors from $\mathcal{M}$ to $\mathcal{N}$ is isomorphic to the category of lax $\mathbf{Vec}$-module functors from $\mathcal{M}$ to $\mathcal{N}$.

Theorems & Definitions (45)

  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 35 more