Biwreaths: a self-contained system in a 2-category that encodes different known algebraic constructions and gives rise to new ones

TL;DR

This paper introduces biwreaths in a 2-category framework, unifying and generalizing various algebraic constructions such as Radford biproducts and Drinfel'd twists, and providing a systematic way to derive known and new algebraic structures.

Contribution

It defines biwreaths as bimonads in a 2-category, connecting known algebraic constructions to this new framework and enabling the discovery of novel algebraic structures through different choices of laws.

Findings

Biwreaths encode known algebraic constructions like Radford biproducts.

The structure laws of biwreaths recover classical algebraic laws such as 2- and 3-(co)cycles.

Different choices within biwreaths lead to new algebraic constructions.

Abstract

We introduce bimonads in a 2-category $\K$ and define biwreaths as bimonads in the 2-category $\bEM(\K)$ of bimonads, in the analogous fashion as Lack and Street defined wreaths. A biwreath is then a system containing a wreath, a cowreath and their mixed versions, but also a 2-cell $\lambda$ in $\bEM(\K)$ governing the compatibility of the monad and the comonad structure of the biwreath. We deduce that the monad laws encode 2-(co)cycles and the comonad laws so called 3-(co)cycles, while the 2-cell conditions of the (co)monad structure 2-cells of the biwreath encode (co)actions twisted by these 2- and 3-(co)cycles. The compatibilities of $\lambda$ deliver concrete expressions of the latter structure 2-cells. We concentrate on the examples of biwreaths in the 2-category induced by a braided monoidal category $\C$ and take for the distributive laws in a biwreath the braidings of the different categories of Yetter-Drinfel'd modules in $\C$. We prove that the before-mentioned properties of a biwreath specified to the latter setting recover on the level of $\C$ different algebraic constructions known in the category ${}_R\M$ of modules over a commutative ring $R$, such as Radford biproduct, Sweedler's crossed product algebra, comodule algebras over a quasi-bialgebra and the Drinfel'd twist. In this way we obtain that the known examples of (mixed) wreaths coming from ${}_R\M$ are not merely examples, rather they are consequences of the structure of a biwreath, and that the form of their structure morphisms originates in the laws inside of a biwreath. Choosing different distributive laws and different 2-cells $\lambda$ in a biwreath, leads to different and possibly new algebraic constructions.