The Fundamental Theorem for weak braided bimonads

TL;DR

This paper extends the fundamental theorem for weak braided bimonads within the framework of monoidal categories, generalizing classical Hopf algebra theory to a broader categorical context with weak Yang-Baxter operators.

Contribution

It develops a generalized fundamental theorem for weak braided bimonads, incorporating weak Yang-Baxter operators in an abstract categorical setting.

Findings

Extended the fundamental theorem to weak braided bimonads

Unified theory of weak (Hopf) bialgebras in monoidal categories

Captured properties of braided (Hopf) bialgebras with weak Yang-Baxter operators

Abstract

The theories of (Hopf) bialgebras and weak (Hopf) bialgebras have been introduced for vector space 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 $\mathbb{A}$, 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 $\tau:HH\to HH$. 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.