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 $L_2 = \struct{S_2, \preceq_2}$:

$\phi: L_1 \to L_2$ is a continuous map:

if and only if:

$\phi$ is a frame homomorphism $\phi: L_2 \to L_1$

Also see

• Category of Locales is Category

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to Locales, $\S 1.1$ Definition (b)