A functorial approach to $n$-abelian categories

Vitor Gulisz

doi:10.5802/crmath.790

Research article - Representation theory

A functorial approach to $n$-abelian categories

Vitor Gulisz ¹

¹ Mathematics Department, Northeastern University, Boston, MA 02115, USA

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1123-1175

Abstract (Original language)
Résumé

We develop a functorial approach to the study of $n$-abelian categories by reformulating their axioms in terms of their categories of finitely presented functors. Such an approach allows the use of classical homological algebra and representation theory techniques to understand higher homological algebra. As an application, we present two possible generalizations of the axioms “every monomorphism is a kernel” and “every epimorphism is a cokernel” of an abelian category to $n$-abelian categories. We also specialize our results to modules over rings, thereby describing when the category of finitely generated projective modules over a ring is $n$-abelian. Moreover, we establish a correspondence for $n$-abelian categories with additive generators, which extends the higher Auslander correspondence.

Nous développons une approche fonctorielle de l’étude des catégories $n$-abéliennes en reformulant leurs axiomes en termes de leurs catégories de foncteurs de présentation finie. Une telle approche permet d’utiliser les techniques de l’algèbre homologique classique et de la théorie des représentations pour comprendre l’algèbre homologique supérieure. Comme application, nous présentons deux généralisations possibles des axiomes « tout monomorphisme est un noyau » et « tout épimorphisme est un conoyau » d’une catégorie abélienne aux catégories $n$-abéliennes. Appliqués au cas des modules sur les anneaux, nos résultats nous permettent également de décrire quand la catégorie des modules projectifs de type fini sur un anneau est $n$-abélienne. De plus, nous établissons une correspondance pour les catégories $n$-abéliennes à générateurs additifs, qui étend la correspondance d’Auslander supérieure.

Received: 2025-05-28
Revised: 2025-08-13
Accepted: 2025-09-05
Published online: 2025-10-13

DOI: 10.5802/crmath.790

Classification: 18A25, 18E99, 18E05, 16E99
Keywords: Functor categories, finitely presented functors, higher homological algebra, $n$-abelian categories
Mots-clés : Catégories de foncteurs, foncteurs de présentation finie, algèbre homologique supérieure, catégories $n$-abéliennes

Author's affiliations:

Vitor Gulisz ¹

¹ Mathematics Department, Northeastern University, Boston, MA 02115, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Vitor Gulisz},
     title = {A functorial approach to $n$-abelian categories},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1123--1175},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.790},
     language = {en},
}

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Vitor Gulisz. A functorial approach to $n$-abelian categories. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1123-1175. doi: 10.5802/crmath.790

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