Geodesic
Categories which are varieties of classical or ordered algebras
Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 39-67

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Following ideas of Lawvere and Linton we prove that classical varieties are precisely the exact categories with a varietal generator. This means a strong generator which is abstractly finite and regularly projective. An analogous characterization of varieties of ordered algebras is also presented. We work with order-enriched categories, and introduce the concept of subcongruence (corresponding to congruence in ordinary categories): it is a relation which is order-reflexive and transitive. Varieties of ordered algebras are precisely the categories with effective subcongruences and a subvarietal generator. This means a strong generator which is abstractly finite and subregularly projective.
Publié le :
Classification : 18C05, 08C05
Keywords: variety, ordered algebras, effective congruences, subcongruence, varietal generator 
Jiří Adámek. Categories which are varieties of classical or ordered algebras. Theory and applications of categories, Lawvere Festschrift, Tome 43 (2025), pp. 39-67. http://geodesic.mathdoc.fr/item/TAC_2025_43_a2/
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     author = {Ji\v{r}{\'\i} Ad\'amek},
     title = {Categories which are varieties of classical or ordered algebras},
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     pages = {39--67},
     year = {2025},
     volume = {43},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2025_43_a2/}
}
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