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

Definition

Locale

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 $A$.

Topological Space

Let $\mathbf{Top}$ denote the category of topological spaces.

Let $\mathbf{KHausTop}$ denote the category of compact Hausdorff spaces.

Let $\tuple{\beta, \iota, \alpha}$ be an adjunction of $\mathbf{KHausTop}$ and $\mathbf{Top}$ where:

$\mathbf{\mathbf{Top}} \to \mathbf{\mathbf{KHausTop}}$ denotes a functor

$\iota: \mathbf{\mathbf{KHausTop}} \to \mathbf{\mathbf{Top}}$ denotes the inclusion functor

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

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

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

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