On bicrossed modules of Hopf algebras

We use Hopf algebroids to formulate a notion of a noncommutative and non-cocommutative Hopf 2-algebra. We show how these arise from a bicrossproduct Hopf algebra with Peiffer identities. In particular, we show that for a Hopf algebra $H$ with bijective antipode, the mirror bicrossproduct Hopf algebra $H▹\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}◀H_{cop}$ is a Hopf 2-algebra. We apply the theorem on Sweedler’s Hopf algebra as an example.