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. This explains conceptually the lifting of (rigid) dualities 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 is build upon a 2-categorical Yang-Baxter operator. We show how this concept, which we call a bilax functor, generalizes many known notions from the theory of Hopf algebras. We propose a 2-category of bilax functors whose 1-cells generalize Yetter-Drinfel?d modules in ordinary categories. We prove that the 2-category of bilax functors from the trivial 2-category is isomorphic to the 2-category of bimonads, and construct a faithful 2-functor from the latter to the 2-category of mixed distributive laws of Power and Watanabe.