A characterisation of extriangulated categories with triangulated structure

TL;DR

This paper characterizes extriangulated categories that can be endowed with a triangulated structure, identifying specific conditions on morphisms involving zero objects.

Contribution

It provides a precise characterization of extriangulated categories that admit a triangulated structure based on morphism properties involving zero objects.

Findings

Categories where 0→X is a deflation for all X

Categories where X→0 is an inflation for all X

Equivalent conditions for triangulated structure presence

Abstract

We give a characterisation of the extriangulated categories which admit the structure of a triangulated category. We show that these are the extriangulated categories where for every object $X$ in the extriangulated category, the morphism $0 \rightarrow X$ is a deflation and the morphism $X \rightarrow 0$ is an inflation.