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 $\tuple{\beta, \iota, \alpha}$ be an adjunction of $\mathbf{KCRegLoc}$ and $\mathbf{Loc}$ where:

$\beta: \mathbf{Loc} \to \mathbf{KCRegLoc}$ denotes a functor

$\iota: \mathbf{KCRegLoc} \to \mathbf{Loc}$ denotes the inclusion functor

Let $\eta:\operatorname{id}_{\mathbf{Loc}} \to \iota \beta$ be the unit of the adjunction $\tuple{\beta, \iota, \alpha}$ where:

$\operatorname{id}_{\mathbf{Loc}}$ denotes the identity functor on $\mathbf{Loc}$

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

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

Also see

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

• Stone-Čech Compactifications of Locale are Unique up to Isomorphism

Sources

Johnstone