# A Note on Actions of a Monoidal Category

## G. Janelidze and G.M. Kelly

An action $* : \cal V \times \cal A \to \cal A$ of a monoidal category $\cal V$ on a category $\cal A$ corresponds to a strong monoidal functor $F : \cal V \to [\cal A,\cal A]$ into the monoidal category of endofunctors of $\cal A$. In many practical cases, the ordinary functor $f : \cal V \to [cal \A, \cal A]$ 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 $\cal V$ and in $[\cal A, \cal A]$, we have an adjunction $Mon F$ left adjoint to $Mon G$ between the category $Mon \cal V$ of monoids in $\cal V$ and the category $Mon [\cal A, \cal A] = Mnd \cal A$ of monads on $\cal A$. 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 $\cal A$ (at least when $\cal V$ is symmetric and closed) into a tensored and cotensored $cal \V$-category ${\bf A}$. We give explicit formulae, as large ends, for the right adjoints $g$ and $Mon G$, and also for some related right adjoints, when they exist; as well as another explicit expression for $Mon G$ as a large limit, which uses a new representation of any (large) limit of monads of two special kinds, and an analogous result for general endofunctors.

Keywords: monoidal category, action, enriched category, monoid, monad, adjunction.

2000 MSC: 18C15, 18D10, 18D20.

Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61-91.

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