#
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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