A Note on Actions of a Monoidal Category

G. Janelidze ; G.M. Kelly

Theory and applications of categories, CT2000, Tome 9 (2000), pp. 61-91.

Résumé

An action $\ast :\mathcal{V}\times \mathcal{A}\to \mathcal{A}$ of a monoidal category $\mathcal{V}$ on a category $\mathcal{A}$ corresponds to a strong monoidal functor $F:\mathcal{V}\to \mathcal{[}\mathcal{A}\mathcal{,}\mathcal{A}]$ into the monoidal category of endofunctors of $\mathcal{A}$. In many practical cases, the ordinary functor $f:\mathcal{V}\to \mathcal{[}\mathcal{c}\mathcal{a}\mathcal{l}\text{A}\mathcal{,}\mathcal{A}]$ 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 $\mathcal{V}$ and in $[\mathcal{A}\mathcal{,}\mathcal{A}]$, we have an adjunction $MonF$ left adjoint to $MonG$ between the category $Mon\mathcal{V}$ of monoids in $\mathcal{V}$ and the category $Mon[\mathcal{A}\mathcal{,}\mathcal{A}]\mathcal{=}\mathcal{M}\mathcal{n}\mathcal{d}\mathcal{A}$ of monads on $\mathcal{A}$. We give sufficient conditions for the existence of the right adjoint $g$, which involve the existence of right adjoints for the functors $X\ast -$ and $\ast A$, and make $\mathcal{A}$ (at least when $\mathcal{V}$ is symmetric and closed) into a tensored and cotensored $cal\text{V}$-category $\mathbf{A}$. 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 (large) limit of monads of two special kinds, and an analogous result for general endofunctors.

Détail

Classification : 18C15, 18D10, 18D20.
Mots-clés : monoidal category, action, enriched category, monoid, monad, adjunction.