Crossed squares of cocommutative Hopf algebras

TL;DR

This paper introduces Hopf crossed squares for cocommutative Hopf algebras, establishing their equivalence with double internal groupoids and internal crossed modules, thus extending classical algebraic structures.

Contribution

It extends the concept of crossed squares to cocommutative Hopf algebras and proves their categorical equivalences with double internal groupoids and crossed modules.

Findings

Hopf crossed squares generalize crossed squares of groups and Lie algebras.

Equivalence established between Hopf crossed squares and double internal groupoids.

Hopf crossed squares are shown to be internal crossed modules in a specific category.

Abstract

In this paper, we define the notion of Hopf crossed square for cocommutative Hopf algebras extending the notions of crossed squares of groups and of Lie algebras. We prove the equivalence between the category of Hopf crossed squares and the category of double internal groupoids in the category of cocommutative Hopf algebras. The Hopf crossed squares turn out to be the internal crossed modules in the category of crossed modules in the category of cocommutative Hopf algebras.