Crossed squares of cocommutative Hopf algebras

In this paper, we define the notion of Hopf crossed squares 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. (C) 2021 Elsevier Inc. All rights reserved.