Hopf Monads: A Survey with New Examples and Applications

TL;DR

This survey explores Hopf monads in monoidal categories, introducing new examples and applications including cross products, analogues of classical theorems, and classifications of related structures.

Contribution

It provides a comprehensive overview of Hopf monads with novel examples, applications, and classifications, expanding the understanding of their role in algebra and category theory.

Findings

New examples of Hopf monads from Galois and Ore extensions

Analogues of fundamental theorems for Hopf algebroids

Classification of Hopf monads on sets and natural numbers

Abstract

We survey the theory of Hopf monads on monoidal categories, and present new examples and applications. As applications, we utilise this machinery to present a new theory of cross products, as well as analogues of the Fundamental Theorem of Hopf algebras and Radford's biproduct Theorem for Hopf algebroids. Additionally, we describe new examples of Hopf monads which arise from Galois and Ore extensions of bialgebras. We also classify Lawvere theories whose corresponding monads on the category of sets and functions become Hopf, as well as Hopf monads on the poset of natural numbers.