nLab Loc

Definition

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

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

$Loc$ is naturally a (1,2)-category; its 2-morphisms are the pointwise ordering of frame homomorphisms.

Definition

The 2-category Locale has

• as objects $X$ frames $Op(X)$;

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

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

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

Properties

For any base topos $S$ the 2-category $Loc(S)$ 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 forms categories of sheaves

$$Sh : Locale \to Topos$$

exhibits $Locale$ 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