nLab
Loc

Definition

$\mathrm{Loc}$ or $\mathrm{Locale}$ is the category whose objects are locales and whose morphisms are continuous maps between locales. By definition, this means that $\mathrm{Loc}$ is the opposite category of Frm, the category of frames.

$\mathrm{Loc}$ is used as a substitute for Top if one wishes to do topology with locales instead of standard topological spaces.

$\mathrm{Loc}$ is naturally a (1,2)-category; its 2-morphism are the pointwise ordering of frame homomorphisms.

Definition

The 2-category Locale has

• as objects $X$ frames $\mathrm{Op}(X)$;

• as morphisms $f:X\to Y$ frame homomorphisms $f^{*}:\mathrm{Op}(Y)\to \mathrm{Op}(X)$;

• a unique 2-morphisms $f\Rightarrow g$ whenever for all $U\in \mathrm{Op}(Y)$ we have a (then necessarily unique) morphism $f^{*}U\to g^{*}U$.

(For instance Johnstone, C1.4, p. 514.)

Properties

For any base topos $E$ the 2-category $\mathrm{Loc}(S)$ of internal locales in $S$ is equivalent to the subcategory of the slice of Topos over $S$ on the localic geometric morphisms. See there for more details.

See locale for more properties.

The 2-functor that formes categories of sheaves

$$\mathrm{Sh}:\mathrm{Locale}\to \mathrm{Topos}$$Sh : Locale \to Topos

exhibits $\mathrm{Locale}$ as a full sub-2-category of Topos. See localic reflection for more on this.

References

For instance Section C1 of

• Peter Johnstone, Sketches of an elephant: a topos theory compendium. .

category: category