Could not include topos theory - contents
nice topological space
nice category of spaces
convenient category of topological spaces
Freudenthal suspension theorem
CW-complex, Hausdorff space, second-countable space, sober space
compact space, paracompact space
connected space, locally connected space, contractible space, locally contractible space
topological vector space, Banach space, Hilbert space
point, real line, plane
sphere, ball, annulus
loop space, path space
Cantor space, Sierpinski space
long line, Warsaw circle
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LocLoc or LocaleLocale is the category whose objects are locales and whose morphisms are continuous maps between locales. By definition, this means that LocLoc is the opposite category of Frm, the category of frames.
LocLoc is used as a substitute for Top if one wishes to do topology with locales instead of standard topological spaces.
LocLoc is naturally a (1,2)-category; its 2-morphism are the pointwise ordering of frame homomorphisms.
The 2-category Locale has
as objects XX frames Op(X)Op(X);
as morphisms f:X→Yf : X \to Y frame homomorphisms f *:Op(Y)→Op(X)f^* : Op(Y) \to Op(X);
a unique 2-morphisms f⇒gf \Rightarrow g whenever for all U∈Op(Y)U \in Op(Y) we have a (then necessarily unique) morphism f *U→g *Uf^* U \to g^* U.
(For instance Johnstone, C1.4, p. 514.)
For any base topos EE the 2-category Loc(S)Loc(S) of internal locales in SS is equivalent to the subcategory of the slice of Topos over SS on the localic geometric morphisms. See there for more details.
See locale for more properties.
The 2-functor that formes categories of sheaves
exhibits LocaleLocale as a full sub-2-category of Topos. See localic reflection for more on this.
For instance Section C1 of