Paired wreaths: towards a 2-categorical atlas of cross products

After we introduced biwreaths and biwreath-like objects in our previous paper, in the present one we define paired wreaths. In a paired wreath there is a monad $B$ and a comonad $F$ over the same 0-cell in a 2-category $\K$, so that $F$ is a left wreath around $B$ and $B$ is a right cowreath around $F$, and moreover, $FB$ is a bimonad in $\K$. The corresponding 1-cell $FB$ in the setting of a biwreath and a biwreath-like object was not necessarilly a bimonad. We obtain a 2-categorical version of the Radford biproduct and Sweedler's crossed (co)product, that are on one hand, both a biwreath and a biwreath-like object, respectively, and on the other hand, they are also paired wreaths. We show that many known crossed (bi)products in the literature are special cases of paired wreaths, including cocycle cross product bialgebras of Bespalov and Drabant in braided monoidal categories. This is a part of a project of constructing a kind of a 2-categorical atlas of all the known crossed (bi)products. We introduce a Hopf datum in $\K$ which contains part of the structure of a paired wreath. We define Yang-Baxter equations and naturality conditions of certain distributive laws in $\K$ and study when Hopf data are paired wreaths. From the notion of a Hopf datum new definitions of Yetter-Drinfel`d modules, (co)module (co)monads, 2-(co)cycles and (co)cycle twisted (co)actions in a 2-categorical setting are suggested.