User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Locales

Definition

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

Let $\mathbf{KCRegLoc}$ denote the category of compact completely regular locales.

Let $i: \mathbf{KCRegLoc} \to \mathbf{Loc}$ be the inclusion functor.

Let $\beta: \mathbf{Loc} \to \mathbf{KCRegLoc}$ be a left adjoint functor of $i$.

For any $A \in \mathbf{Loc}$:

$\beta A$ is called a Stone-Čech Compactification of $A$.

Also see

• Inclusion from Compact Completely Regular Locales into Locales has Adjoint Functor

• Stone-Čech Compactifications of Locale are Unique upto Isomorphism

Sources

Johnstone