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Abelian quotients of categories of $n$-exangles

Yutong Zhou

TL;DR

The paper addresses constructing abelian quotients from categories of $n$-exangles in an $n$-exangulated category $(al C,,rak{s})$ by translating $n$-exangle morphisms into module-theoretic data. It uses the morphism-category framework to identify $S(al C)/al R_2$ with module categories under projective or injective hypotheses, and proves abelian structure when $n$ is even. The main contributions include explicit equivalences $S(al C)/al R_2 \u2261 ext{mod-}(al C/[al P])$ and, dually, $( ext{mod-}((al C/[al I])^{op}))^{op}$, a detailed description of projectives via $P_A^ullet$, a characterization of simple objects as Auslander–Reiten $n$-exangles, and natural dualities connecting the two module-theoretic realizations. These results extend known abelian quotient constructions from triangulated and exact/extriangulated settings to the broader $n$-exangulated framework, offering a unified perspective on higher homological structures.

Abstract

The notion of $n$-exangulated categories was introduced by Herschend-Liu-Nakaoka, which is a simultaneous generalization of $n$-exact categories in the sense of Jasso and $(n+2)$-angulated categories in the sense of Geiss-Kelier-Oppermann. Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ a full subcategory of $\mathscr{C}$ containing $\mathcal{P}$. In the present paper, It is proved that a certian quotient category of $\mathfrak{s}$-def$(\mathcal{M})$ is abelian. We denoted by $S(\mathscr{C})$ the category of $n$-exangles, whose object are given by distinguished $n$-exangles in $\mathscr{C}$. If $\mathcal{M}=\mathscr{C}$, we obtain that a certain ideal quotient category $S(\mathscr{C})/\mathcal{R}_2$ is equivalent to the category of finitely presented modules mod-$(\mathscr{C}/[\mathcal{P}])$. Furthermore, we present the quotient category $S(\mathscr{C})/\mathcal{R}_2$ always has an abelian structure when taking $n$ as an even number. The abelian quotient $S(\mathscr{C})/\mathcal{R}_2$ admits some nice properties. We describe the projective objects in $S(\mathscr{C})/\mathcal{R}_2$ and characterize the simple objects in $S(\mathscr{C})/\mathcal{R}_2$ as Auslander-Reiten $n$-exangle sequences in $\mathscr{C}$.

Abelian quotients of categories of $n$-exangles

TL;DR

The paper addresses constructing abelian quotients from categories of -exangles in an -exangulated category by translating -exangle morphisms into module-theoretic data. It uses the morphism-category framework to identify with module categories under projective or injective hypotheses, and proves abelian structure when is even. The main contributions include explicit equivalences and, dually, , a detailed description of projectives via , a characterization of simple objects as Auslander–Reiten -exangles, and natural dualities connecting the two module-theoretic realizations. These results extend known abelian quotient constructions from triangulated and exact/extriangulated settings to the broader -exangulated framework, offering a unified perspective on higher homological structures.

Abstract

The notion of -exangulated categories was introduced by Herschend-Liu-Nakaoka, which is a simultaneous generalization of -exact categories in the sense of Jasso and -angulated categories in the sense of Geiss-Kelier-Oppermann. Let be an -exangulated category with enough projectives and a full subcategory of containing . In the present paper, It is proved that a certian quotient category of -def is abelian. We denoted by the category of -exangles, whose object are given by distinguished -exangles in . If , we obtain that a certain ideal quotient category is equivalent to the category of finitely presented modules mod-. Furthermore, we present the quotient category always has an abelian structure when taking as an even number. The abelian quotient admits some nice properties. We describe the projective objects in and characterize the simple objects in as Auslander-Reiten -exangle sequences in .

Paper Structure

This paper contains 6 sections, 16 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Morphism Categories
  4. $n$-Exangulated Categories
  5. Identifying quotients of morphism categories as module categories
  6. Abelian quotients of categories of $n$-exangles

Key Result

Lemma 2.12

LZ Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category, and \xymatrix{ X_0 \ar[r]^-{d^X_0} & X_1 \ar[r]^-{d^X_1} & \cdots \ar[r]^-{d^X_{n-2}}&X_{n-1} \ar[r]^-{d^X_{n-1}}& X_n \ar[r]^-{d^X_{n}} & X_{n+1} \ar@{-->}[r]^{\delta}&}be a distinguished $n$-exangle. Then the following In particular, $d^X_i$ is a weak cokernel of $d^X_{i-1}$ and $d^X_{i-1}$ is a weak kernel of $d^X_i

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 23 more