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

##### 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)