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Internal object actions. (English) Zbl 1121.18004
Summary: We describe the place, among other known categorical constructions, of the internal object actions involved in the categorical notion of semidirect product, and introduce a new notion of representable action providing a common categorical description for the automorphism group of a group, for the algebra of derivations of a Lie algebra, and for the actor of a crossed module.

MSC:
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18C20 Eilenberg-Moore and Kleisli constructions for monads
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
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