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In [[mathematics]], especially in [[order theory]], a '''complete Heyting algebra''' is a [[Heyting algebra]] which is [[completeness (order theory)|complete]] as a [[lattice (order)|lattice]]. Complete Heyting algebras are the [[object (category theory)|objects]] of three different [[category (category theory)|categories]]; the category '''CHey''', the category '''Loc''' of locales, and its [[opposite (category theory)|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.

== 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 {{nowrap|1=( ''x'' &and; &minus; )}} has a [[Adjoint functors|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 (order theory)|distributivity]] law holds:
: <math>x \wedge \bigvee_{s \in S} s = \bigvee_{s \in S} (x \wedge s).</math>
* ''P'' is a distributive lattice, i.e., for all ''x'', ''y'' and ''z'' in ''P'', we have
: <math>x \wedge ( y \vee z ) = ( x \wedge y ) \vee ( x \wedge z )</math>

: and the meet operations {{nowrap|1=( ''x'' &and; &minus; )}} are [[Scott continuous]] for all ''x'' in ''P'' (i.e., preserve the suprema of [[directed set]]s) .

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

== Frames and locales ==
The [[object (category theory)|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 [[monotonic|monotone]]) functions that [[limit preserving function (order theory)|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 [[dual (category theory)|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
:<math>f\colon X\to Y</math>
be any map. The [[power set]]s ''P''(''X'') and ''P''(''Y'') are [[complete Boolean algebra]]s, and the map
: <math>f^{-1}\colon P(Y)\to P(X)</math>
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
: <math>f^{-1}\colon O(Y)\to O(X)</math>
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
: <math>f\colon X\to Y</math>
to the map
: <math>O(f)\colon O(X)\to O(Y)</math>
in '''Loc''' that is defined in '''Frm''' to be the inverse image frame homomorphism
: <math>f^{-1}\colon O(Y)\to O(X).</math>
It is common, given a map of locales
: <math>f\colon A\to B</math>
in '''Loc''', to write
: <math>f^{*}\colon B\to A</math>
for the frame homomorphism that defines it in '''Frm'''. Hence, using this notation, ''O''(''ƒ'') is defined by the equation {{nowrap|1=''O''(''&fnof;'')<sup>*</sup> = ''&fnof;''<sup>&minus;1</sup>.}}

Conversely, any locale ''A'' has a topological space ''S''(''A'') that best approximates the locale, called its ''spectrum''. In addition, any map of locales
: <math>f\colon A\to B</math>
determines a continuous map
: <math>S(A)\to S(B),</math>
and this assignment is functorial: letting ''P''(1) denote the locale that is obtained as the powerset of the terminal set {{nowrap|1=1 = { * },}} the points of ''S''(''A'') are the maps
: <math>p\colon P(1)\to A</math>
in '''Loc''', i.e., the frame homomorphisms
: <math>p^*\colon A\to P(1).</math>
For each {{nowrap|1=''a'' &isin; ''A''}} we define the set {{nowrap|1=''U<sub>a''</sub> &sube; ''S''(''A'')}} that consists of the points {{nowrap|1=''p'' &isin; ''S''(''A'')}} such that {{nowrap|1=''p''*(''a'') = { * }.}} It is easy to verify that this defines a frame homomorphism {{nowrap|1=''A'' &rarr; ''P''(''S''(''A'')),}} whose image is therefore a topology on ''S''(''A''). Then, if
: <math>f\colon A\to B</math> is a map of locales,
to each point {{nowrap|1=''p'' &isin; ''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
: <math>S(f)\colon S(A)\to S(B).</math>

This defines a functor <math>S</math> 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.

* {{cite book | zbl=1034.18001 | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 }}

[[Category:Order theory]]
[[Category:Algebraic structures]]