A functorial approach to $n$-abelian categories

Abstract

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.

arXiv comment62 pages. v4: revised version, accepted for publication in C. R. Math. Acad. Sci. Paris

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• $n$-angulated categories
• $n$-abelian and $n$-exact categories
• Classifying exact categories via Wakamatsu tilting
• Classifications of exact structures and Cohen-Macaulay-finite algebras
• Higher Auslander correspondence for dualizing $R$-varieties
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• Higher Auslander correspondence for exact categories
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Cited by in corpus (1)

• On the equivalence between the existence of $n$-kernels and $n$-cokernels