Complete Heyting algebra
Encyclopedia
In mathematics
, especially in order theory
, a complete Heyting algebra is a Heyting algebra
which 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 morphism
s, and thus get distinct names. Only the morphisms of CHey are homomorphism
s 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.
(P, ≤) that is a complete lattice
. Then P is a complete Heyting algebra if any of the following equivalent conditions hold:
ordered by inclusion is a complete Heyting algebra.
.
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 algebra
s, and the map
is a homomorphism of complete Boolean algebras. Suppose the spaces X and Y are topological space
s, endowed with the topology O(X) and O(Y) of open set
s 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
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 the points of S(A) are the maps
in Loc, i.e., the frame homomorphisms
For each we define the set that consists of the points such that It is easy to verify that this defines a frame homomorphism whose image is therefore a topology on S(A). Then, if
to each point 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
, a complete Heyting algebra is a Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
which is complete
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
as a lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
. 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 morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s, and thus get distinct names. Only the morphisms of CHey are homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s of complete Heyting algebras.
Locales and frames form the foundation of pointless topology
Pointless topology
In mathematics, pointless topology is an approach to topology that avoids mentioning points. The name 'pointless topology' is due to John von Neumann...
, which, instead of building on point-set topology, recasts the ideas of general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
in categorical terms, as statements on frames and locales.
Definition
Consider a partially ordered setPartially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
(P, ≤) that is a complete lattice
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum . Complete lattices appear in many applications in mathematics and computer science...
. Then P is a complete Heyting algebra if any of the following equivalent conditions hold:
- P is a Heyting algebra, i.e. the operation has a right adjointAdjoint functorsIn mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
(also called the lower adjoint of a (monotone) Galois connectionGalois connectionIn mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
), for each element x of P. - For all elements x of P and all subsets S of P, the following infinite distributivityDistributivity (order theory)In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima...
law holds:
-
- P is a distributive lattice, i.e., for all x, y and z in P, we have
-
- and P is meet continuous, i.e. the meet operations are Scott continuous for all x in P.
Examples
The system of all open sets of a given topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
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 morphismMorphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
.
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
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...
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 algebra
Complete Boolean algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum . Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing...
s, and the map
is a homomorphism of complete Boolean algebras. Suppose the spaces X and Y are topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s, endowed with the topology O(X) and O(Y) of open set
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
s 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
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
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
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 the points of S(A) are the maps
in Loc, i.e., the frame homomorphisms
-
.
For each we define the set that consists of the points such that It is easy to verify that this defines a frame homomorphism whose image is therefore a topology on S(A). Then, if
-
is a map of locales,
to each point 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
Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...
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 PressCambridge University PressCambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by Henry VIII in 1534, it is the world's oldest publishing house, and the second largest university press in the world...
, 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.