Gabriel-Quillen embedding for $n$-exact categories

TL;DR

This paper extends the Gabriel-Quillen embedding theorem to $n$-exact categories, providing new insights into their structure and examples, and exploring ways to realize them as $n$-cluster tilting subcategories.

Contribution

It introduces an analog of the Gabriel-Quillen embedding theorem for $n$-exact categories and discusses methods to realize these categories as $n$-cluster tilting subcategories.

Findings

Provided an embedding theorem for $n$-exact categories.

Constructed an example of an $n$-exact category not being an $n$-cluster tilting subcategory.

Suggested two approaches to realize $n$-exact categories as $n$-cluster tilting subcategories.

Abstract

Our first aim is to provide an analog of the Gabriel-Quillen embedding theorem for $n$-exact categories. Also we give an example of an $n$-exact category that is not an $n$-cluster tilting subcategory, and we suggest two possible ways for realizing $n$-exact categories as $n$-cluster tilting subcategory.