Frames And Locales
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, especially in
order theory Order theory is a branch of mathematics that investigates the 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 intr ...
, a complete Heyting algebra is a
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
that is
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
as a
lattice Lattice may refer to: Arts and design * Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material * Lattice (music), an organized grid model of pitch ratios * Lattice (pastry), an orna ...
. Complete Heyting algebras are the
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
of three different
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being *Categories (Aristotle), ''Categories'' (Aristotle) *Category (Kant) ...
; 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 In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s, and thus get distinct names. Only the morphisms of CHey are
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s of complete Heyting algebras. Locales and frames form the foundation of
pointless topology In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this appr ...
, which, instead of building on
point-set topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, recasts the ideas of
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
in categorical terms, as statements on frames and locales.


Definition

Consider a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
(''P'', ≤) that is a
complete lattice In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
. Then ''P'' is a complete Heyting algebra or frame if any of the following equivalent conditions hold: * ''P'' is a Heyting algebra, i.e. the operation (x\land\cdot) has a
right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...
(also called the lower adjoint of a (monotone)
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
), for each element ''x'' of ''P''. * For all elements ''x'' of ''P'' and all subsets ''S'' of ''P'', the following infinite
distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, ...
law holds: ::x \land \bigvee_ s = \bigvee_ (x \land s). * ''P'' is a distributive lattice, i.e., for all ''x'', ''y'' and ''z'' in ''P'', we have ::x \land ( y \lor z ) = ( x \land y ) \lor ( x \land z ) : and the meet operations (x\land\cdot) are Scott continuous (i.e., preserve the suprema of
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
s) for all ''x'' in ''P''. The entailed definition of Heyting implication is a\to b=\bigvee\. Using a bit more category theory, we can equivalently define a frame to be a
cocomplete In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one ...
cartesian closed In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in math ...
poset In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
.


Examples

The system of all open sets of a given
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
ordered by inclusion is a complete Heyting algebra.


Frames and locales

The
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
of the category CHey, the category Frm of frames and the category Loc of locales are complete Heyting algebras. These categories differ in what constitutes a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
: * The morphisms of Frm are (necessarily
monotone Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonic ...
) functions that
preserve The word preserve may refer to: Common uses * Fruit preserves, a type of sweet spread or condiment * Nature reserve, an area of importance for wildlife, flora, fauna or other special interest, usually protected Arts, entertainment, and media ...
finite meets and arbitrary joins. * The definition of Heyting algebras crucially involves the existence of right adjoints to the binary meet operation, which together define an additional
implication operation Implication may refer to: Logic * Logical consequence (also entailment or logical implication), the relationship between statements that holds true when one logically "follows from" one or more others * Material conditional (also material conse ...
. Thus, the morphisms of CHey are morphisms 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 f: X\to Y be any map. The
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
s ''P''(''X'') and ''P''(''Y'') are
complete Boolean algebra In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolea ...
s, and the map f^: P(Y)\to P(X) is a homomorphism of complete Boolean algebras. Suppose the spaces ''X'' and ''Y'' are
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s, endowed with the topology ''O''(''X'') and ''O''(''Y'') of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s on ''X'' and ''Y''. Note that ''O''(''X'') and ''O''(''Y'') are subframes of ''P''(''X'') and ''P''(''Y''). If f is a continuous function, then f^: O(Y)\to O(X) preserves finite meets and arbitrary joins of these subframes. This shows that ''O'' is a
functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from the category Top of topological spaces to Loc, taking any continuous map : f: X\to Y to the map : O(f): O(X)\to O(Y) in Loc that is defined in Frm to be the inverse image frame homomorphism : f^: O(Y)\to O(X). Given a map of locales f: A\to B in Loc, it is common to write f^*: B\to A for the frame homomorphism that defines it in Frm. Using this notation, O(f) is defined by the equation O(f)^* = f^. Conversely, any locale ''A'' has a topological space ''S''(''A''), called its ''spectrum'', that best approximates the locale. In addition, any map of locales f: A\to B determines a continuous map S(A)\to S(B). Moreover this assignment is functorial: letting ''P''(1) denote the locale that is obtained as the power set of the terminal set 1=\, the points of ''S''(''A'') are the maps p: P(1)\to A in Loc, i.e., the frame homomorphisms p^*: A\to P(1). For each a\in A we define U_a as the set of points p\in S(A) such that p^*(a) =\. It is easy to verify that this defines a frame homomorphism A\to P(S(A)), whose image is therefore a topology on ''S''(''A''). Then, if f: A\to B is a map of locales, to each point p\in S(A) we assign the point S(f)(q) defined by letting S(f)(p)^* be the composition of p^* with f^*, hence obtaining a continuous map S(f): S(A)\to S(B). This defines a functor S 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 In cryptography, SOBER is a family of stream ciphers initially designed by Greg Rose of QUALCOMM Australia starting in 1997. The name is a contrived acronym for ''S''eventeen ''O''ctet ''B''yte ''E''nabled ''R''egister. Initially the cipher wa ...
''. The adjunction between topological spaces and locales restricts to an
equivalence of categories In category theory, a branch of abstract 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 fro ...
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 University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
, Cambridge, 1982. () : ''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 Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, ...
, ''Continuous Lattices and Domains'', In ''Encyclopedia of Mathematics and its Applications'', Vol. 93, Cambridge University Press, 2003. : ''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, . *


External links

* {{nlab, id=locale, title=Locale Order theory Algebraic structures