User:Leigh.Samphier/Topology/Definition:Category of Locales
Definition
The category of locales, denoted $\mathbf{Loc}$, is the dual category of the category of frames.
Locale
An object of $\mathbf{Loc}$ is called a locale.
That is, a locale is a complete lattice $\struct {L, \preceq}$ satisfying the infinite join distributive law:
| \(\ds \forall a \in L, S \subseteq L:\) | \(\ds a \wedge \bigvee S = \bigvee \set {a \wedge s : S \in S} \) |
where $\bigvee S$ denotes the supremum $\sup S$.
Continuous Map
A morphism of $\mathbf{Loc}$ is called a continuous map.
That is, for locales $L_1 = \struct{S_1, \preceq_1}$ and $L_2 = \struct{S_2, \preceq_2}$:
- $f: L_1 \to L_2$ is a continuous map:
- there exists a frame homomorphism $\phi: L_2 \to L_1$ such that $f = \phi^{\operatorname{op}}$
Category of Locales with Localic Mappings
From User:Leigh.Samphier/Topology/Frame Homomorphism is Lower Adjoint of Unique Galois Connection, every frame homomorphism is the lower adjoint of a unique Galois connection.
From All Infima Preserving Mapping is Upper Adjoint of Galois Connection, every continuous map is uniqely determined by the localic mapping that is the upper adjoint of the frame homomorphism that corresponds to the continuous map.
This, together with the unintuitive notion of a continuous map as the dual morphism of a frame homomorphism $\phi : L_2 \to L_1$, leads some sources to define the continuous map $f : L_1 \to L_2$ to be the localic mapping $f : L_1 \to L_2$. This definition of a continuous map results in the category of locales with localic mappings.
From User:Leigh.Samphier/Topology/Category of Locales with Localic Mappings is Isomorphic to Category of Locales we have that the category of locales with localic mappings and the category of locales are isomorphic categories but they are not identical.
Also see
Sources
- 1982: Peter T. Johnstone: Stone Spaces: Chapter II: Introduction to Locales, $\S 1.1$
- 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter II: Frames and Locales. Spectra, $\S 2.1$