Definition:Continuous Map (Locale)

Definition

Let $\mathbf{Loc}$ denote the category of locales.

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

• Definition:Frame Homomorphism

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter II: Introduction to Locales, $\S1.1$ Definition (b)