Hopf crossed module (co)algebras

Abstract

Given a crossed module $\chi$, we introduce Hopf $\chi$-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of representations are monoidal and $\chi$-graded (meaning that both objects and morphisms have degrees which are related via $\chi$).