# Complete Heyting algebra

In mathematics, especially in order theory, a **complete Heyting algebra** is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category **CHey**, the category **Loc** of locales, and its opposite, the category **Frm** of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of **CHey** are homomorphisms of complete Heyting algebras.

Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales.

## Definition

Consider a partially ordered set (*P*, ≤) that is a complete lattice. Then *P* is a *complete Heyting algebra* if any of the following equivalent conditions hold:

*P*is a Heyting algebra, i.e. the operation (*x*∧ − ) has a right adjoint (also called the lower adjoint of a (monotone) Galois connection), for each element*x*of*P*.- For all elements
*x*of*P*and all subsets*S*of*P*, the following infinite distributivity law holds:

*P*is a distributive lattice, i.e., for all*x*,*y*and*z*in*P*, we have

- and the meet operations (
*x*∧ − ) are Scott continuous for all*x*in*P*(i.e., preserve the suprema of directed sets) .

## Examples

The system of all open sets of a given topological space ordered by inclusion is a complete Heyting algebra.

## Frames and locales

The objects of the category **CHey**, the category **Frm** of frames and the category **Loc** of locales are the complete lattices satisfying the infinite distributive law. These categories differ in what constitutes a morphism.

The morphisms of **Frm** are (necessarily monotone) functions that preserve finite meets and arbitrary joins. Such functions are not homomorphisms of complete Heyting algebras. The definition of Heyting algebras crucially involves the existence of right adjoints to the binary meet operation, which together define an additional implication operation ⇒. Thus, a *homomorphism of complete Heyting algebras* is a morphism of frames that in addition preserves implication. The morphisms of **Loc** are opposite to those of **Frm**, and they are usually called maps (of locales).

The relation of locales and their maps to topological spaces and continuous functions may be seen as follows. Let

be any map. The power sets *P*(*X*) and *P*(*Y*) are complete Boolean algebras, and the map

is a homomorphism of complete Boolean algebras. Suppose the spaces *X* and *Y* are topological spaces, endowed with the topology *O*(*X*) and *O*(*Y*) of open sets on *X* and *Y*. Note that *O*(*X*) and *O*(*Y*) are subframes of *P*(*X*) and *P*(*Y*). If *ƒ* is a continuous function, then

preserves finite meets and arbitrary joins of these subframes. This shows that *O* is a functor from the category **Top** of topological spaces to the category **Loc** of locales, taking any continuous map

to the map

in **Loc** that is defined in **Frm** to be the inverse image frame homomorphism

It is common, given a map of locales

in **Loc**, to write

for the frame homomorphism that defines it in **Frm**. Hence, using this notation, *O*(*ƒ*) is defined by the equation *O*(*ƒ*)^{*} = *ƒ*^{−1}.

Conversely, any locale *A* has a topological space *S*(*A*) that best approximates the locale, called its *spectrum*. In addition, any map of locales

determines a continuous map

and this assignment is functorial: letting *P*(1) denote the locale that is obtained as the powerset of the terminal set 1 = { * }, the points of *S*(*A*) are the maps

in **Loc**, i.e., the frame homomorphisms

For each *a* ∈ *A* we define the set *U _{a}* ⊆

*S*(

*A*) that consists of the points

*p*∈

*S*(

*A*) such that

*p**(

*a*) = { * }. It is easy to verify that this defines a frame homomorphism

*A*→

*P*(

*S*(

*A*)), whose image is therefore a topology on

*S*(

*A*). Then, if

to each point *p* ∈ *S*(*A*) we assign the point *S*(*ƒ*)(*q*) defined by letting *S*(*ƒ*)(p)* be the composition of *p** with *ƒ**, hence obtaining a continuous map

This defines a functor from **Loc** to **Top**, which is right adjoint to *O*.

Any locale that is isomorphic to the topology of its spectrum is called *spatial*, and any topological space that is homeomorphic to the spectrum of its locale of open sets is called *sober*. The adjunction between topological spaces and locales restricts to an equivalence of categories between sober spaces and spatial locales.

Any function that preserves all joins (and hence any frame homomorphism) has a right adjoint, and, conversely, any function that preserves all meets has a left adjoint. Hence, the category **Loc** is isomorphic to the category whose objects are the frames and whose morphisms are the meet preserving functions whose left adjoints preserve finite meets. This is often regarded as a representation of **Loc**, but it should not be confused with **Loc** itself, whose morphisms are formally the same as frame homomorphisms in the opposite direction.

## Literature

- P. T. Johnstone,
*Stone Spaces*, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press, Cambridge, 1982. (ISBN 0-521-23893-5)

*Still a great resource on locales and complete Heyting algebras.*

- G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott,
*Continuous Lattices and Domains*, In*Encyclopedia of Mathematics and its Applications*, Vol. 93, Cambridge University Press, 2003. ISBN 0-521-80338-1

*Includes the characterization in terms of meet continuity.*

- Francis Borceux:
*Handbook of Categorical Algebra III*, volume 52 of*Encyclopedia of Mathematics and its Applications*. Cambridge University Press, 1994.

*Surprisingly extensive resource on locales and Heyting algebras. Takes a more categorical viewpoint.*

- Steven Vickers,
*Topology via logic*, Cambridge University Press, 1989, ISBN 0-521-36062-5.

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