nLab
Loc

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Topology

Contents

Definition

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.

Definition

The 2-category Locale has

  • as objects XX frames Op(X)Op(X);

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

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

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

Properties

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

Sh:LocaleTopos Sh : Locale \to Topos

exhibits LocaleLocale as a full sub-2-category of Topos. See localic reflection for more on this.

References

For instance Section C1 of

category: category

Revised on July 25, 2011 20:46:31 by Urs Schreiber (82.113.99.13)