The Fundamental Theorem for weak braided bimonads

The theories of (Hopf) bialgebras and weak (Hopf) bialgebras have been introduced for vector spantr, categories over fields and make heavily use of the tensor product. As first generalisations, these notions were formulated for monoidal categories, with braidings if needed. The present authors developed a theory of bimonads and Hopf monads H on arbitrary categories A, employing distributive laws, allowing for a general form of the Fundamental Theorem for Hopf algebras. For tau-bimonads H, properties of braided (Hopf) bialgebras were captured by requiring a Yang Baxter operator tau : HH -> HH. The purpose of this paper is to extend the features of weak (Hopf) bialgebras to this general setting including an appropriate form of the Fundamental Theorem. This subsumes the theory of braided Hopf algebras (based on weak Yang Baxter operators) as considered by Alonso Alvarez and others. (C) 2017 Elsevier Inc. All rights reserved.