Hopf measuring comonoids and enrichment: HOPF MEASURING COMONOIDS AND ENRICHMENT

We study the existence of universal measuring comonoids P(A, B) for a pair of monoids A, B in a braided monoidal closed category, and the associated enrichment of the category of monoids over the monoidal category of comonoids. In symmetric categories, we show that if A is a bimonoid and B is a commutative monoid, then P(A, B) is a bimonoid; in addition, if A is a cocommutative Hopf monoid then P(A, B) always is Hopf. If A is a Hopf monoid, not necessarily cocommutative, then P(A, B) is Hopf if the fundamental theorem of comodules holds; to prove this we give an alternative description of the dualizable P(A, B)-comodules and use the theory of Hopf (co) monads. We explore the examples of universal measuring comonoids in vector spaces and graded spaces.