The long $n$-exact sequence theorem in $n$-abelian categories

TL;DR

This paper proves that the long $n$-exact sequence theorem is valid in any $n$-abelian category and applies it to establish an $n$-abelian version of Wakamatsu's Lemma.

Contribution

It demonstrates the validity of the long $n$-exact sequence theorem in all $n$-abelian categories and introduces an $n$-abelian analogue of Wakamatsu's Lemma.

Findings

Long $n$-exact sequence theorem holds in any $n$-abelian category.

An $n$-abelian version of Wakamatsu's Lemma is established.

The results extend classical homological algebra to the $n$-abelian setting.

Abstract

In this note, we show that the long $n$-exact sequence theorem holds in any $n$-abelian category. As an application, the $n$-abelian version of Wakamatsu's Lemma is given.