From right (n+2)-angulated categories to n-exangulated categories

TL;DR

This paper explores the relationship between right (n+2)-angulated categories and n-exangulated categories, establishing conditions under which one structure induces the other and analyzing the role of semi-equivalence and trivial inflations.

Contribution

It introduces the notion of right semi-equivalence in right (n+2)-angulated categories and characterizes when an n-exangulated category can be viewed as a right (n+2)-angulated category.

Findings

The standard right (n+2)-angulated category modulo a strongly covariantly finite subcategory is right semi-equivalence.

An n-exangulated category has a right (n+2)-angulated structure with right semi-equivalence iff the suspension functor is right semi-equivalence.

An n-exangulated category admits a right (n+2)-angulated structure with right semi-equivalence iff all morphisms to zero are trivial inflations.

Abstract

The notion of right semi-equivalence in a right $(n+2)$-angulated category is defined in this article. Let $\mathscr C$ be an $n$-exangulated category and $\mathscr X$ is a strongly covariantly finite subcategory of $\mathscr C$. We prove that the standard right $(n+2)$-angulated category $\mathscr C/\mathscr X$ is right semi-equivalence under a natural assumption. As an application, we show that a right $(n+2)$-angulated category has an $n$-exangulated structure if and only if the suspension functor is right semi-equivalence. Besides, we also prove that an $n$-exangulated category $\mathscr C$ has the structure of a right $(n+2)$-angulated category with right semi-equivalence if and only if for any object $X\in\mathscr C$, the morphism $X\to 0$ is a trivial inflation.