Esakia duality

In mathematics, '''Esakia duality''' is a theorem establishing a correspondence between Heyting algebras and certain ordered topological spaces called Esakia spaces; more precisely, it states that the category of Heyting algebras is dually equivalent to the category of Esakia spaces.

Let '''Esa''' denote the category of Esakia spaces and Esakia morphisms.

Let {{math|''H''}} be a Heyting algebra, {{math|''X''}} denote the set of prime filters of {{math|''H''}}, and {{math|≤}} denote set-theoretic inclusion on the prime filters of {{math|''H''}}. Also, for each {{math|''a'' ∈ ''H''}}, let {{math|''φ''(''a'') {{=}} {''x'' ∈ ''X'' : ''a'' ∈ ''x''}}}, and let {{math|''τ''}} denote the topology on {{math|''X''}} generated by {{math| {''φ''(''a''), ''X'' − ''φ''(''a'') : ''a'' ∈ ''H''}}}.

Theorem:<ref name=":0">{{cite journal|last=Esakia|first=Leo|date=1974|title=Topological Kripke models|journal=Soviet Math|volume=15|issue=1|pages=147–151}}</ref> {{math|(''X'', ''τ'', ≤)}} is an Esakia space, called the ''Esakia dual'' of {{math|''H''}}. Moreover, {{math|''φ''}} is a Heyting algebra isomorphism from {{math|''H''}} onto the Heyting algebra of all clopen up-sets of {{math|(''X'',''τ'',≤)}}. Furthermore, each Esakia space is isomorphic in '''Esa''' to the Esakia dual of some Heyting algebra.

This representation of Heyting algebras by means of Esakia spaces is functorial and yields a dual equivalence between the categories * '''HA''' of Heyting algebras and Heyting algebra homomorphisms and * '''Esa''' of Esakia spaces and Esakia morphisms.

Theorem:<ref name=":0" /><ref>{{cite journal|last=Esakia|first=L|date=1985|title=Heyting Algebras I. Duality Theory|journal=Metsniereba, Tbilisi}}</ref><ref>{{cite book|url=https://pure.uva.nl/ws/files/3836071/40465_Bezhanishvili.pdf|title=Lattices of intermediate and cylindric modal logics|last=Bezhanishvili|first=N.|date=2006|publisher=Amsterdam Institute for Logic, Language and Computation (ILLC)|isbn=978-90-5776-147-8}}</ref> '''HA''' is dually equivalent to '''Esa'''.

The duality can also be expressed in terms of spectral spaces, where it says that the category of Heyting algebras is dually equivalent to the category of Heyting spaces.<ref>see section 8.3 in * {{cite book | last1=Dickmann | first1=Max | last2=Schwartz | first2= Niels | last3=Tressl | first3= Marcus | title=Spectral Spaces| doi=10.1017/9781316543870 | year=2019 | publisher=Cambridge University Press | series=New Mathematical Monographs | volume=35 | location=Cambridge | isbn=9781107146723 }} </ref>

==References== {{reflist}}

{{DEFAULTSORT:Esakia Duality}} Category:Topology Category:Lattice theory Category:Duality (mathematics)