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abelian sheaf cohomology
model structure on simplicial presheaves
higher topos theory
(∞,1)-sheaf, ∞-stack, derived stack
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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 objects X frames Op(X);
as morphisms f:X→Y frame homomorphisms f *:Op(Y)→Op(X);
a unique 2-morphisms f⇒g whenever for all U∈Op(Y) we have a (then necessarily unique) morphism f *U→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
Sh : Locale \to Topos
exhibits Locale as a full sub-2-category of Topos. See localic reflection for more on this.
For instance Section C1 of