Idempotent completion of extriangulated categories

TL;DR

This paper proves that the idempotent completion of an extriangulated category naturally inherits an extriangulated structure, enabling the transfer of cotorsion pairs and recollement properties.

Contribution

It establishes that idempotent completion preserves the extriangulated structure and extends key categorical properties to this completion.

Findings

Idempotent completion admits a natural extriangulated structure.

Cotorsion pairs induce similar pairs in the completion under certain conditions.

Recollement structure is preserved in the idempotent completion.

Abstract

Extriangulated categories were introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. In this paper, we show that the idempotent completion of an extriangulated category admits a natural extriangulated structure. As applications, we prove that cotorsion pairs in an extriangulated category induce cotorsion pairs in its idempotent completion under certain condition, and the idempotent completion of a recollement of extriangulated categories is still a recollement.