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We introduce bimonads in a 2-category $\K$ and define biwreaths as bimonadsin the 2-category $\bEM(\K)$ of bimonads, in the analogous fashion as Lack andStreet defined wreaths. A biwreath is then a system containing a wreath, acowreath and their mixed versions, but also a 2-cell $\lambda$ in $\bEM(\K)$governing the compatibility of the monad and the comonad structure of thebiwreath. We deduce that the monad laws encode 2-(co)cycles and the comonadlaws so called 3-(co)cycles, while the 2-cell conditions of the (co)monadstructure 2-cells of the biwreath encode (co)actions twisted by these 2- and3-(co)cycles. The compatibilities of $\lambda$ deliver concrete expressions ofthe latter structure 2-cells. We concentrate on the examples of biwreaths inthe 2-category induced by a braided monoidal category $\C$ and take for thedistributive laws in a biwreath the braidings of the different categories ofYetter-Drinfel'd modules in $\C$. We prove that the before-mentioned propertiesof a biwreath specified to the latter setting recover on the level of $\C$different algebraic constructions known in the category ${}_R\M$ of modulesover a commutative ring $R$, such as Radford biproduct, Sweedler's crossedproduct algebra, comodule algebras over a quasi-bialgebra and the Drinfel'dtwist. In this way we obtain that the known examples of (mixed) wreaths comingfrom ${}_R\M$ are not merely examples, rather they are consequences of thestructure of a biwreath, and that the form of their structure morphismsoriginates in the laws inside of a biwreath. Choosing different distributivelaws and different 2-cells $\lambda$ in a biwreath, leads to different andpossibly new algebraic constructions.

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