The Karoubi envelope and weak idempotent completion of an extriangulated category

Dixy Msapato

doi:10.1007/s10485-021-09664-8

Abstract

We show that the idempotent completion and weak idempotent completion of an extriangulated category are also extriangulated.

Highlights

Extriangulated categories were introduced by Nakaoka and Palu in [13] as a simultaneous generalisation of exact categories and triangulated categories in the context of the study of cotorsion pairs
There are many extriangulated categories which a neither exact nor triangulated. It is shown in [8, Theorem 2.4] that the category of Cohen-Macaulay differential graded modules over certain Gorenstein differential graded algebras is extriangulated. Another is the subcategory K [−1,0](proj ), which is the subcategory of complexes concetrated in degree -1 and degree 0 in K b(proj ), where is an Artin algebra; see [15, Proposition 4.39]
We have shown that the idempotent completion Cis an extriangulated category

Summary

Introduction

Extriangulated categories were introduced by Nakaoka and Palu in [13] as a simultaneous generalisation of exact categories and triangulated categories in the context of the study of cotorsion pairs. For any morphism (a, c) : δ → δ of E-extensions, there exists b ∈ C(B, B ) such that the following diagram commutes. There exists an object E in C, a commutative diagram f f in C and an E-extension δ ∈ E(E, A) realised by the sequence A −→ h C −h→ E, which satisfy the following compatibilities: (i) s(( f )∗δ ) = [D −→ d E −→ e F].

Results

Conclusion