On the category of Poisson Hopf algebras

A Poisson Hopf algebra is both a Poisson algebra and a Hopf algebra such that the comultiplication and
the counit are morphisms of Poisson algebras. Such objects are situated at the border between Poisson
geometry and quantum groups.  We investigate the category of Poisson Hopf algebras with respect to
properties such as (co)completeness or existence of (co)free objects. For instance, we prove that there
exists a free Poisson Hopf algebra on any coalgebra or, equivalently that the forgetful functor from the
category of Poisson Hopf algebras to the category of coalgebras has a left adjoint. In particular, we also
show that the category of Poisson Hopf algebras is a reflective subcategory of the category of Poisson
bialgebras. Along the way, we point out the differences between the category of Poisson Hopf algebras
and the category of Hopf algebras.