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An object $P$ in a monoidal category $\mathcal{C}$ is called pivotal if itsleft dual and right dual objects are isomorphic. Given such an object and achoice of dual $Q$, we construct the category $\mathcal{C}(P,Q)$, of objectswhich intertwine with $P$ and $Q$ in a compatible manner. We show that thiscategory lifts the monoidal structure of $\mathcal{C}$ and the closed structureof $\mathcal{C}$, when $\mathcal{C}$ is closed. If $\mathcal{C}$ has suitablecolimits we show that $\mathcal{C}(P,Q)$ is monadic and thereby construct afamily of Hopf monads on arbitrary closed monoidal categories $\mathcal{C}$. Wealso introduce the pivotal cover of a monoidal category and extend our work toarbitrary pivotal diagrams.

Pivotal Objects in Monoidal Categories and Their Hopf Monads