A note on actions of a monoidal category. (English) Zbl 1009.18005

Janelidze, G.; Kelly, G. M.

Theory Appl. Categ. 9, 61-91 (2001).

An action of a monoidal category \({\mathcal V}\) on a category \({\mathcal A}\) can be viewed as a strong monoidal functor \(F\) from \({\mathcal V}\) to the endofunctor category of \({\mathcal A}\). The paper deals with the question of a right adjoint \(G\) for \(F\) and the induced adjunction between monoids in \({\mathcal V}\) and monoids on \({\mathcal A}\). There is a new representation of any monad as a large limit of endomorphism-like monads.

Reviewer: Ross H.Street (North Ryde)

MSC:

18D15	Closed categories (closed monoidal and Cartesian closed categories, etc.)
18C15	Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18D10	Monoidal, symmetric monoidal and braided categories (MSC2010)
18D20	Enriched categories (over closed or monoidal categories)

Keywords:

monad; enriched category; monoidal category