algebra universalis

, Volume 49, Issue 1, pp 35–49 | Cite as

On the duality between varieties and algebraic theories

  • J. Adámek
  • F.W. Lawvere
  • J. Rosický
Regular article


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.

2000 Mathematics Subject Classification: 18C10, 08B99.¶Key words and phrases: Variety, algebraic theory, duality. 


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Copyright information

© Birkhäuser Verlag Basel, 2003

Authors and Affiliations

  • J. Adámek
    • 1
  • F.W. Lawvere
    • 2
  • J. Rosický
    • 3
  1. 1.Technical University Braunschweig, Postfach 3329, 38023 Braunschweig, Germany, e-mail: adamek@iti.cs.tu-bs.deDE
  2. 2.The State University of New York at Buffalo, Diefendorf Hall, Buffalo, NY 14214, USA, e-mail: wlawvere@acsu.bu.alo.eduUS
  3. 3.Masaryk University, Janáčkovo nám. 2a, 662 95 Brno, Czech Republic, e-mail: rosicky@math.muni.czCZ

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