Divisible, Torsion-Free, and Act Regular Generalized Act Wreath Products

We construct generalized act wreath products over wreath products of a monoid with a small category. This construction generalizes wreath products of acts over wreath products of monoids, arbitrary acts over their endomorphism monoids, and graphs as acts over their monoids of strong endomorphisms. We characterize divisible, torsion-free, and act regular generalized act wreath products and apply the results in particular to prove the respective endoproperties for projective acts.