User:Leigh.Samphier/Topology/Definition:Locale (Lattice Theory)
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Definition
Let $\mathbf{Loc}$ denote the category of locales.
An object of $\mathbf{Loc}$ is called a locale.
That is, a locale is a complete lattice $\struct {L, \preceq}$ satisfying the infinite join distributive law:
| \(\ds \forall a \in L, S \subseteq L:\) | \(\ds a \wedge \bigvee S = \bigvee \set {a \wedge s : S \in S} \) |
where $\bigvee S$ denotes the supremum $\sup S$.
Frames vs Locales vs Complete Heyting Algebras
If we are only concerned with category theoretic objects, the terms frame and locale and complete Heyting algebra are synonymous. (See Characterization of Locale)
It is only when we consider the associated morphisms that they become different:
- For frames the associated morphisms are frame homomorphisms. (See Definition:Category of Frames).
- For locales the associated morphisms are continuous maps. (See Definition:Category of Locales).
- For complete Heyting algebra the associated morphisms are homomorphisms that also preserve the relative pseudocomplement operation $\to$. (See Heyting homomorphism).
Also see
Sources
- 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text {II}$: Introduction to Locales, $\S 1.1$
- 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter $2$: Locales and Localic Maps, $\S 2.2$