Milnor–Moore categories and monadic decomposition

In this paper monoidal Hom-Lie algebras, Lie color algebras, Lie superalgebras and other type of generalized Lie algebras are recovered by means of an iterated construction, known as monadic decomposition of functors, which is based on Eilenberg–Moore categories. To this aim we introduce the notion of Milnor–Moore category as a monoidal category for which a Milnor–Moore type Theorem holds. We also show how to lift the property of being a Milnor–Moore category whenever a suitable monoidal functor is given and we apply this technique to provide examples.