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(Co)algebraic pearls

DOI: 10.4230/LIPIcs.CALCO.2021.6

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

Authors: Jiří Adámek and Jiří Rosický

Published in: LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

Abstract

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.

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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)

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@InProceedings{adamek_et_al:LIPIcs.CALCO.2021.6,
  author =	{Ad\'{a}mek, Ji\v{r}{\'\i} and Rosick\'{y}, Ji\v{r}{\'\i}},
  title =	{{Which Categories Are Varieties?}},
  booktitle =	{9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)},
  pages =	{6:1--6:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-212-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{211},
  editor =	{Gadducci, Fabio and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.6},
  URN =		{urn:nbn:de:0030-drops-153610},
  doi =		{10.4230/LIPIcs.CALCO.2021.6},
  annote =	{Keywords: variety, many-sorted algebra, abstractly finite object, effective object, strong generator}
}

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

Abstract

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