Affine sets: The structure of complete objects and duality

An existence theorem for completions of categories of T 0 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.