In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Esakia duality is the
dual equivalence between the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
of
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'' call ...
s and the category of
Esakia spaces. Esakia duality provides an order-topological representation of Heyting algebras via Esakia spaces.
Let Esa denote the category of Esakia spaces and
Esakia morphisms.
Let be a Heyting algebra, denote the set of
prime filters of , and denote set-theoretic inclusion on the prime filters of . Also, for each , let , and let denote the topology on generated by .
Theorem:
is an Esakia space, called the ''Esakia dual'' of . Moreover, is a Heyting algebra
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
from onto the Heyting algebra of all
clopen
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of and are antonyms, but their mathematical def ...
up-sets of . 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
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" ...
and
* Esa of Esakia spaces and Esakia morphisms.
Theorem:
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.
[see section 8.3 in * ]
See also
*
Duality theory for distributive lattices In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of distributive lattice, bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This du ...
References
{{DEFAULTSORT:Esakia Duality
Topology
Lattice theory
Duality theories