Affine sets: The structure of complete objects and duality

An existence theorem for completions of categories of T0 objects of some kind of topological categories over Set is given, and an internal characterization of complete objects in these categories is established. As a consequence, we recover the existence of completions in several categories studied in topology (such us closure spaces, α-spaces, topological spaces, approach spaces and fuzzy spaces) together with descriptions of their complete objects. A Duality Theorem is also provided, rendering many familiar dualities (e.g., Stone duality, Tarski duality) “internal” dualities.