# nLab Loc

topos theory

## Theorems

#### Topology

topology

algebraic topology

# Contents

## 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-morphism 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-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.)

## Properties

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.

## References

For instance Section C1 of

category: category

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