Higher Auslander's defect and classifying substructures of n-exangulated categories

TL;DR

This paper extends Auslander's defect and Auslander-Reiten duality to n-exangulated categories, providing a framework to classify substructures via defect categories, thus generalizing several existing higher categorical concepts.

Contribution

It introduces an n-exangulated version of Auslander's defect and duality, and classifies substructures using defect categories in n-exangulated categories.

Findings

Established an n-exangulated version of Auslander's defect.

Derived an Auslander-Reiten duality formula for n-exangulated categories.

Provided a classification of substructures using defect categories.

Abstract

Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories and $(n+2)$-angulated categories. In this article, we give an $n$-exangulated version of Auslander's defect and Auslander-Reiten duality formula. Moreover, we also give a classification of substructures (=closed subbifunctors) of a given skeletally small $n$-exangulated category by using the category of defects.