  We introduce Hopf polyads in order to unify Hopf monads and group actions on
monoidal categories. A polyad is a lax functor from a small category (its
source) to the bicategory of categories, and a Hopf polyad is a comonoidal
polyad whose fusion operators are invertible. The main result states that the
lift of a Hopf polyad is a strong (co)monoidal action-type polyad (or strong
monoidal pseudofunctor). The lift of a polyad is a new polyad having simpler
structure but the same category of modules. We show that, under certain
assumptions, a Hopf polyad can be `wrapped up' into a Hopf monad. This
generalizes the fact that finite group actions on tensor categories can be seen
as Hopf monads. Hopf categories in the sense of Batista, Caenepeel and
Vercruysse can be viewed as Hopf polyads in a braided setting via the notion of
Hopf polyalgebras. As a special case of the main theorem, we generalize a
description of the center of graded fusion category due to Turaev and
Virelizier to tensor categories: if $C$ is a $G$-graded (locally bounded)
tensor category, then $G$ acts on the relative center of $C$ with respect to
the degree one part $C_1$, and the equivariantization of this action is the
center of $C$.
