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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.

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)
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