Limits and colimits of Hopf algebras

Abstract

It is shown that for any commutative unital ring R the category Hopf R of R-Hopf algebras is locally presentable and a coreflective subcategory of the category Bialg R of R-bialgebras, admitting cofree Hopf algebras over arbitrary R-algebras. The proofs are based on an explicit analysis of the construction of colimits of Hopf algebras, which generalizes an observation of Takeuchi. Essentially be a duality argument also the dual statement, namely that Hopf R is closed in Bialg R under limits, is shown to hold, provided that the ring R is von Neumann regular. It then follows that Hopf R is reflective in Bialg R and admits free Hopf algebras over arbitrary R-coalgebras, for any von Neumann regular ring R. Finally, Takeuchi's free Hopf algebra construction is analysed and shown to be simply a composition of standard categorical constructions. By simple dualization also a construction of the Hopf coreflection is provided.