Category of Locales is Category

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Theorem

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

Then:

$\mathbf{Loc}$ is a category

Proof

By definition, the category of locales is:

the dual category of the category of frames

From Category of Frames is Category:

the category of frames is a category

From Dual Category is Category:

the category of locales is a category

$\blacksquare$

Sources

• 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to Locales, $\S 1.1$ Definition (b)