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

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.