Monads and comonads in module categories

TL;DR

This paper explores the relationships between monads and comonads in module categories, revealing new insights into algebraic structures like corings, bialgebras, and Hopf algebras through categorical analysis.

Contribution

It investigates categories of modules and comodules associated with Hom functors, providing new characterizations of corings, bialgebras, and Hopf algebras using categorical frameworks.

Findings

Categories of C-comodules and Hom_C-modules are equivalent if C is coseparable.

A bialgebra H is a Hopf algebra iff Hom_R(H,-) forms a Hopf bimonad.

Categories of H-Hopf modules and bimodules are equivalent to module categories.

Abstract

Let $A$ be a ring and $\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\otimes_A B: \M_A\to \M_A$ is a monad (or triple). Similarly, an $A $-bimodule $\C$ is an $A$-coring provided the functor $-\otimes_A\C:\M_A\to \M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\otimes_A B$ and comodules (or coalgebras) of $-\otimes_A\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\Hom_A(B,-)$ and $\Hom_A(\C,-)$ are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of $\Hom_A(B,-)$-comodules is isomorphic to the category of $B$-modules, while the category of $\Hom_A(\C,-)$-modules (called $\C$-contramodules by Eilenberg and Moore) need not be equivalent to the category of $\C$-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of $\C$-comodules and $\Hom_A(\C,-)$-modules are equivalent provided $\C$ is a coseparable coring. Furthermore, a bialgebra $H$ over a commutative ring $R$ is a Hopf algebra if and only if $\Hom_R(H-)$ is a Hopf bimonad on $\M_R$ and in this case the categories of $H$-Hopf modules and mixed $\Hom_R(H,-)$-bimodules are both equivalent to $\M_R$.