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.
Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61-91 http://www.tac.mta.ca/tac/volumes/9/n4/n4.dvi http://www.tac.mta.ca/tac/volumes/9/n4/n4.ps http://www.tac.mta.ca/tac/volumes/9/n4/n4.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n4/n4.dvi ftp://ftp.tac.mta.ca/pub/tac/html/volumes/9/n4/n4.psTAC Home