User:Leigh.Samphier/Topology/Category of Locales with Localic Mappings is Category

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Theorem

Let $\mathbf{Loc_*}$ denote the category of locales with localic mappings.

Then:

$\mathbf{Loc_*}$ is a metacategory

Proof

Let us verify the axioms $(\text C 1)$ up to $(\text C 3)$ for a metacategory.

For any two localic mappings their composition (in the usual set theoretic sense) is again a localic mapping by User:Leigh.Samphier/Topology/Composite Localic Mapping is Localic Mapping.

For any locale $L = \struct{S, \preceq}$, we have the identity mapping $\operatorname{id}_S$.

From User:Leigh.Samphier/Topology/Identity Mapping is Localic Mapping we have $\operatorname{id}_S$ is a localic mapping.

By Identity Mapping is Left Identity and Identity Mapping is Right Identity, this is the identity morphism for $L$.

Finally by Composition of Mappings is Associative, the associative property is satisfied.

Hence $\mathbf{Loc_*}$ is a metacategory.

$\blacksquare$

Sources

• 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter II: Frames and Locales. Spectra, $\S 2.2$