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Hereditary $n$-exangulated categories

Jian He, Jing He, Panyue Zhou

Abstract

Herschend-Liu-Nakaoka introduced the concept of $n$-exangulated categories as higher-dimensional analogues of extriangulated categories defined by Nakaoka-Palu. The class of $n$-exangulated categories contains $n$-exact categories and $(n+2)$-angulated categories as specific examples. In this article, we introduce the notion of hereditary $n$-exangulated categories, which generalize hereditary extriangulated categories. We provide two classes of hereditary $n$-exangulated categories through closed subfunctors. Additionally, we define the concept of $0$-Auslander $n$-exangulated categories and discuss the circumstances under which these two classes of hereditary $n$-exangulated categories become $0$-Auslander.

Hereditary $n$-exangulated categories

Abstract

Herschend-Liu-Nakaoka introduced the concept of -exangulated categories as higher-dimensional analogues of extriangulated categories defined by Nakaoka-Palu. The class of -exangulated categories contains -exact categories and -angulated categories as specific examples. In this article, we introduce the notion of hereditary -exangulated categories, which generalize hereditary extriangulated categories. We provide two classes of hereditary -exangulated categories through closed subfunctors. Additionally, we define the concept of -Auslander -exangulated categories and discuss the circumstances under which these two classes of hereditary -exangulated categories become -Auslander.
Paper Structure (4 sections, 8 theorems, 53 equations)

This paper contains 4 sections, 8 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hereditary $n$-exangulated categories
  4. $0$-Auslander $n$-exangulated categories

Key Result

Theorem 1.1

(see Theorem main1 for details) Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category and $\mathscr{X}$ be a cluster-tilting subcategory of $\mathscr{C}$. We define Then $(\mathscr{C},\mathbb{E}_{\mathscr{X}},\mathfrak{s}\space\mid_{\mathbb{E}_{\mathscr{X}}})$ and $(\mathscr{C},\mathbb{E}^{\mathscr{X}},\mathfrak{s}\space\mid_{\mathbb{E}^{\mathscr{X}}})$ are two hereditary $n$

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 16 more