Some Properties and Some Problems on Set Functors

We study properties of functors on categories of sets (classes) together with set (class) functions. In particular, we investigate the notion of inclusion preserving functor, and we discuss various monotonicity and continuity properties of set functors. As a consequence of these properties, we show that some classes of set operators do not admit functorial extensions. Then, starting from Aczel's Special Final Coalgebra Theorem, we study the class of functors uniform on maps, we present and discuss various examples of functors which are not uniform on maps but still inclusion preserving, and we discuss simple characterization theorems of final coalgebras as fixpoints. We present a number of conjectures and problems.