Could not include topos theory - contents
CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
$Loc$ 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-morphism are the pointwise ordering of frame homomorphisms.
The 2-category Locale has
as morphisms $f : X \to Y$ frame homomorphisms $f^* : Op(Y) \to Op(X)$;
a unique 2-morphisms $f \Rightarrow g$ 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.)
For any base topos $E$ 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 formes categories of sheaves
exhibits $Locale$ as a full sub-2-category of Topos. See localic reflection for more on this.
For instance Section C1 of