Right triangulated categories: As extriangulated categories, aisles and co-aisles

TL;DR

This paper explores right triangulated categories, providing intrinsic characterizations for when they possess extriangulated structures and relate to (co-)aisles of (co-)t-structures within larger triangulated categories.

Contribution

It offers new criteria for identifying extriangulated structures in right triangulated categories and clarifies their role as (co-)aisles in associated t-structures.

Findings

Characterization of extriangulated structures in right triangulated categories

Conditions for these categories to appear as (co-)aisles of (co-)t-structures

Insight into the relationship between right triangulated and triangulated categories

Abstract

Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories have a natural extriangulated structure and are appearing as the (co-)aisle of a (co-)t-structure in an associated triangulated category.