Categorical centers and Yetter--Drinfel`d-modules as 2-categorical (bi)lax structures

The bicategorical point of view provides a natural setting for many concepts in the representation theory of monoidal categories. We show that centers of twisted bimodule categories correspond to categories of 2-dimensional natural transformations and modifications between the deloopings of the twisting functors. We also show that dualities lift to centers of twisted bimodule categories. Inspired by the notion of (pre)bimonoidal functors due to McCurdy and Street and by bilax functors of Aguiar and Mahajan, we study 2-dimensional functors which are simultaneously lax and colax with a compatibility condition. Our approach uses a sort of 2-categorical Yang-Baxter operators, but the idea could equally be carried out using a kind of 2-categorical braidings. We show how this concept, which we call bilax functors, generalize many known notions from the theory of Hopf algebras. We propose a 2-category of bilax functors whose 1-cells generalize the notions of Yetter-Drinfel`d modules in ordinary categories, and a type of bimonads and mixed distributive laws in 2-categories. We show that the 2-category of bilax functors from the trivial 2-category is isomorphic to the 2-category of bimonads, and that there is a faithful 2-functor from the latter to the 2-category of mixed distributive laws of Power and Watanabe.