# nLab Loc

topos theory

## Theorems

#### Topology

topology

algebraic topology

# Contents

## Definition

$\mathrm{Loc}$ or $\mathrm{Locale}$ is the category whose objects are locales and whose morphisms are continuous maps between locales. By definition, this means that $\mathrm{Loc}$ is the opposite category of Frm, the category of frames.

$\mathrm{Loc}$ is used as a substitute for Top if one wishes to do topology with locales instead of standard topological spaces.

$\mathrm{Loc}$ 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 $\mathrm{Op}\left(X\right)$;

• as morphisms $f:X\to Y$ frame homomorphisms ${f}^{*}:\mathrm{Op}\left(Y\right)\to \mathrm{Op}\left(X\right)$;

• a unique 2-morphisms $f⇒g$ whenever for all $U\in \mathrm{Op}\left(Y\right)$ 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 $\mathrm{Loc}\left(S\right)$ 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

$\mathrm{Sh}:\mathrm{Locale}\to \mathrm{Topos}$Sh : Locale \to Topos

exhibits $\mathrm{Locale}$ 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)