SET-VALUED FUNCTORS ARE EQUIVALENT TO CATEGORY ACTIONS

ABSTRACT: A category is a generalization of a monoid and a functor is a generalization of a monoid homomorphism. Hence a set-valued functor can be considered as a generalization of a representation of a monoid by transformations of a set. In monoid case such representations are equivalent to acts over monoids. We show that, analogously, the category of contravariant set-valued functors is naturally equivalent to a category of right category actions.