On Subcategories of the Category of Hopf Algebras

The various canonical subcategories of the category HopfR of Hopf algebras over a commutative ring R, like those of (co)commutative Hopf algebras or Hopf algebras whose antipode is bijective or of order 2, are shown to be locally presentable categories and reflective and coreflective in their respective supercategories. The reflectivity results provided only hold for commutative von Neumann regular rings, while most of the coreflectivity results are valid over any ring. As a consequence one gets existence of free commutative Hopf algebras over coalgebras and cofree cocommutative Hopf algebras over algebras.