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).
Read more about this topic: Galois Connection
Famous quotes containing the word connections:
“The quickness with which all the “stuff” from childhood can reduce adult siblings to kids again underscores the strong and complex connections between brothers and sisters.... It doesn’t seem to matter how much time has elapsed or how far we’ve traveled. Our brothers and sisters bring us face to face with our former selves and remind us how intricately bound up we are in each other’s lives.”
—Jane Mersky Leder (20th century)