On the duality between varieties and algebraic theories

Abstract

Every variety \( \mathcal{V} \) of finitary algebras is known to have an essentially unique algebraic theory \( Th (\mathcal{V}) \) which is Cauchy complete, i.e., all idempotents split in \( Th (\mathcal{V}) \). This defines a duality between varieties (and algebraically exact functors) and Cauchy complete theories (and theory morphisms). Algebraically exact functors are defined as the right adjoints preserving filtered colimits and regular epimorphisms; or, more succintly: as the functors preserving limits and sifted colimits.