  In this paper we focus on a certain self-distributive multiplication on
coalgebras, which leads to so-called rack bialgebra. Inspired by semi-group
theory (adapting the Suschkewitsch theorem), we do some structure theory for
rack bialgebras and cocommutative Hopf dialgebras. We also construct canonical
rack bialgebras (some kind of enveloping algebras) for any Leibniz algebra and
compare to the existing constructions.
  We are motivated by a differential geometric procedure which we call the
Serre functor:
  To a pointed differentible manifold with multiplication is associated its
distribution space supported in the chosen point. For Lie groups, it is
well-known that this leads to the universal enveloping algebra of the Lie
algebra. For Lie racks, we get rack-bialgebras, for Lie digroups, we obtain
cocommutative Hopf dialgebras.
