It is a classic result in lattice theory that a poset is a complete lattice i↵ it can be realized as fixpoints of a closure operator on a powerset. Dragalin [9,10] observed that a poset is a locale (complete Heyting algebra) i↵ it can be realized as fixpoints of a nucleus on the locale of upsets of a poset. He also showed how to generate a nucleus on upsets by adding a structure of “paths” to a...

Based on original work in [TV03], this paper gives a representation theorem for the Scott continuous maps between frames in terms of natural transformations between functors indexed by frames. The result specializes to frame homomorphisms, thereby giving a representation theorem for all locale maps (i.e. continuous maps as defined in locale theory).