  De Vries duality yields a dual equivalence between the category of compact
Hausdorff spaces and a category of complete Boolean algebras with a proximity
relation on them, known as de Vries algebras. We extend de Vries duality to
completely regular spaces by replacing the category of de Vries algebras with
certain extensions of de Vries algebras. This is done by first formulating a
duality between compactifications and de Vries extensions, and then
specializing to the extensions that correspond to Stone-\v{C}ech
compactifications.
