A pair of objects (P,Q) in a monoidal category C, is called a pivotal pair if there exist a family of duality morphisms, making Q both a left dual and a right dual of P. We introduce the correct notion of morphisms between such pairs, and thereby define the pivotal cover of a monoidal category. Given such a pair (P,Q), we construct the category C(P,Q), of objects which intertwine with P and Q in a compatible manner and show that C(P,Q) lifts the monoidal structure of C as well as the closed structure of C, when C is closed. If C has suitable colimits, we construct a family of Hopf monads which correspond to such pairs in C and present the resulting families of braided Hopf algebras and Hopf algebroids, when C is a braided category or the category of bimodules over a base algebra, respectively.
Keywords: Monoidal category, closed category, pivotal category, Hopf monad, Hopf algebra, tensor category
2020 MSC: 18M05, 18D15, 16T99, 18C20
Theory and Applications of Categories, Vol. 37, 2021, No. 11, pp 287-325.