Monoidal ring and coring structures obtained from wreaths and cowreaths

TL;DR

This paper explores how wreaths and cowreaths in a monoidal category give rise to algebraic structures like rings and corings on modules, unifying various constructions such as smash products.

Contribution

It establishes a categorical framework linking wreaths and cowreaths to (co)ring structures on modules, generalizing existing algebraic constructions.

Findings

Wreaths correspond to ring structures on modules.

Cowreaths correspond to coring structures on modules.

Several known smash product constructions are special cases of this framework.

Abstract

Let $A$ be an algebra in a monoidal category $\Cc$, and let $X$ be an object in $\Cc$. We study $A$-(co)ring structures on the left $A$-module $A\ot X$. These correspond to (co)algebra structures in $EM(\Cc)(A)$, the Eilenberg-Moore category associated to $\Cc$ and $A$. The ring structures are in bijective correspondence to wreaths in $\Cc$, and their category of representations is the category of representations over the induced wreath product. The coring structures are in bijective correspondence to cowreaths in $\Cc$, and their category of corepresentations is the category of generalized entwined modules. We present several examples coming from (co)actions of Hopf algebras and their generalizations. Various notions of smash products that have appeared in the literature appear as special cases of our construction.