Every variety ν of finitary algebras is known to have an essentially unique algebraic theory Th(ν) which is Cauchy complete, i.e., all idempotents split in Th(ν). 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.
CITATION STYLE
Adámek, J., Lawvere, F. W., & Rosický, J. (2003). On the duality between varieties and algebraic theories. Algebra Universalis, 49(1), 35–49. https://doi.org/10.1007/s000120300002
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