Definition:Category of Locales
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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 $L_2 = \struct{S_2, \preceq_2}$:
- $\phi: L_1 \to L_2$ is a continuous map:
- $\phi$ is a frame homomorphism $\phi: L_2 \to L_1$
Also see
Sources
- 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to Locales, $\S 1.1$ Definition (b)