Which Categories Are Varieties?

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Authors

ADÁMEK Jiří ROSICKÝ Jiří

Year of publication 2021
Type Article in Proceedings
Conference CALCO 2021: 9th Conference on Algebra and Coalgebra in Computer Science
MU Faculty or unit

Faculty of Science

Citation
Web https://drops.dagstuhl.de/opus/volltexte/2021/15361/
Doi http://dx.doi.org/10.4230/LIPIcs.CALCO.2021.6
Keywords variety; many-sorted algebra; abstractly finite object; effective object; strong generator
Description 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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