Given a monoid in a monoidal category, there is an induced monad obtained by tensoring with from the right. Consider the Kleisli category of the monad . It is not necessarily a monoidal category; there is a need for an additional structure (distributive law of a sort) in order to make such a lift. If the monoidal category in question is the monoidal category of left modules over a Hopf algebra , then any commutative algebra in the center provides a canonical lift to the Kleisli category
Dynamical extension or a dynamization of a monoidal category is an abstract framework to aid the construction of so called dynamical quantum groups which are in turn related to dynamical quantum Yang-Baxter equation.
In Donin-Mudrov (2006) a relation to bialgebroids has been exhibited.