Galois Connection - Galois Connections As Morphisms

Galois Connections As Morphisms

Galois connections also provide an interesting class of mappings between posets which can be used to obtain categories of posets. Especially, it is possible to compose Galois connections: given Galois connections (f ∗, f _∗) between posets A and B and (g ∗, g _∗) between B and C, the composite (g ∗f ∗, f _∗g _∗) is also a Galois connection. When considering categories of complete lattices, this can be simplified to considering just mappings preserving all suprema (or, alternatively, infima). Mapping complete lattices to their duals, this categories display auto duality, that are quite fundamental for obtaining other duality theorems. More special kinds of morphisms that induce adjoint mappings in the other direction are the morphisms usually considered for frames (or locales).