On Finitary Functors

A simple criterion for a functor to be finitary is presented: we call F finitely bounded if for all objects X every finitely generated subobject of F X factorizes through the F -image of a finitely generated subobject of X.This is equivalent to F being finitary for all functors between 'reasonable' locally finitely presentable categories, provided that F preserves monomorphisms.We also discuss the question when that last assumption can be dropped.The answer is affirmative for functors between categories such as Set, K-Vec (vector spaces), boolean algebras, and actions of any finite group either on Set or on K-Vec for fields K of characteristic 0.All this generalizes to locally λ-presentable categories, λ-accessible functors and λpresentable algebras.As an application we obtain an easy proof that the Hausdorff functor on the category of complete metric spaces is ℵ 1 -accessible.