Which Categories Are Varieties? ((Co)algebraic pearls)

Authors Jiří Adámek, Jiří Rosický

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Author Details

Jiří Adámek
  • Department of Mathematics, Czech Technical University in Prague, Czech Republic
  • Institute of Theoretical Computer Science, Technische Universität Braunschweig, Germany
Jiří Rosický
  • Department of Mathematics and Statistics, Faculty of Sciences, Masaryk University, Brno, Czech Republic

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Jiří Adámek and Jiří Rosický. Which Categories Are Varieties? ((Co)algebraic pearls). In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Categories equivalent to single-sorted varieties of finitary algebras were characterized in the famous dissertation of Lawvere. We present a new proof of a slightly sharpened version: those are precisely the categories with kernel pairs and reflexive coequalizers having an abstractly finite, effective strong generator. A completely analogous result is proved for varieties of many-sorted algebras provided that there are only finitely many sorts. In case of infinitely many sorts a slightly weaker result is presented: instead of being abstractly finite, the generator is required to consist of finitely presentable objects.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic language theory
  • variety
  • many-sorted algebra
  • abstractly finite object
  • effective object
  • strong generator


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