A Note on Actions of a Monoidal Category

An action ∗ : V × A−→ A of a monoidal category V on a category A corresponds to a strong monoidal functor F : V−→ [A,A] into the monoidal category of endofunctors of A. In many practical cases, the ordinary functor f : V−→ [A,A] underlying the monoidal F has a right adjoint g; and when this is so, F itself has a right adjoint G as a monoidal functor—so that, passing to the categories of monoids (also called “algebras”) in V and in [A,A], we have an adjunction MonF MonG between the category MonV of monoids in V and the category Mon[A,A] = MndA of monads on A. We give sufficient conditions for the existence of the right adjoint g, which involve the existence of right adjoints for the functors X∗ – and – ∗A, and make A (at least when V is symmetric and closed) into a tensored and cotensored V-category A. We give explicit formulae, as large ends, for the right adjoints g and MonG, and also for some related right adjoints, when they exist; as well as another explicit expression for MonG as a large limit, which uses a new representation of any monad as a (large) limit of monads of two special kinds, and an analogous result for general endofunctors.