$n$-Extension closed subcategories of $n$-exangulated categories

TL;DR

This paper explores how subcategories of $n$-exangulated and $n$-exact categories inherit structures, establishes a strong Obscure Axiom, and characterizes $n$-exact categories within $n$-exangulated categories.

Contribution

It demonstrates that $n$-extension closed subcategories naturally inherit $n$-exangulated structures and characterizes $n$-exact categories via $n$-exangulated categories with specific morphism properties.

Findings

Subcategories inherit $n$-exangulated structures through restriction.

A strong Obscure Axiom holds for $n$-exangulated categories with $n \,\geq\, 2$.

Characterization of $n$-exact categories within $n$-exangulated categories.

Abstract

Let $n$ be a positive integer. We show that an $n$-extension closed subcategory of an $n$-exangulated category naturally inherits an $n$-exangulated structure through restriction of the ambient $n$-exangulated structure. Furthermore, we show that a strong version of the Obscure Axiom holds for $n$-exangulated categories, where $n \geq 2$. This allows us to characterize $n$-exact categories as $n$-exangulated categories with monic inflations and epic deflations. We also show that for an extriangulated category condition (WIC), which was introduced by Nakaoka and Palu, is equivalent to the underlying additive category being weakly idempotent complete. We then apply our results to show that $n$-extension closed subcategories of an $n$-exact category are again $n$-exact. Furthermore, we recover and improve results of Klapproth and Zhou.