On Hopf 2-algebras

Our main goal in this paper is to translate the diagram below relating groups, Lie algebras, and Hopf algebras to the corresponding 2-objects, that is, to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras, and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans [Higher-dimensional algebra. VI. Lie 2-algebras. Theory and Applications of Categories 12 (2004): 492–538.], another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.