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)