Dualities for universal (co)acting Hopf monoids

In general, universal (co) measuring (co) monoids and universal (co) acting bi/Hopf monoids, which prove to be a useful tool in the classification of quantum symmetries, do not always exist. In order to ensure their existence, the support of a given object was recently introduced in AGV3 and used to restrict the class of objects considered when defining universal (co) acting objects. It is well-known that, in contrast with the universal coacting Hopf algebra, for actions on algebras over a field it is usually difficult to describe the universal acting Hopf algebra explicitly and this turns the duality theorem into an important investigation tool. In the present paper we establish duality results for universal (co) measuring (co) monoids and universal (co) acting bi/Hopf monoids in pre-rigid braided monoidal categories C. In addition, when the base category C is closed monoidal, we provide a convenient uniform approach to the aforementioned universal objects in terms of the cosupports, which in this case become subobjects of internal hom-objects. In order to explain our constructions, we use the language of locally initial objects. Known results from the literature are recovered when the base category is the category of vector spaces over a field. New cases where our results can be applied are explored, including categories of (co) modules over (co) quasitriangular Hopf algebras, Yetter-Drinfeld modules and dg-vector spaces.