Pointlessly - Categories of Frames and Locales

Categories of Frames and Locales

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i.e. every (even infinite) subset {a_i} of L has a supremum ⋁a_i such that

for all b in L. These frames, together with lattice homomorphisms that respect arbitrary suprema, form a category. The dual of the category of frames is called the category of locales and generalizes the category Top of all topological spaces with continuous functions. The consideration of the dual category is motivated by the fact that every continuous map between topological spaces X and Y induces a map between the lattices of open sets in the opposite direction as for every continuous function f: X → Y and every open set O in Y the inverse image f -1(O) is an open set in X.