The left (right) triangulated structures of the stable categories

TL;DR

This paper extends the construction of left and right triangulated structures on stable categories to more general settings and explores their applications in abelian model categories.

Contribution

It generalizes previous results on triangulated structures of stable categories and connects these structures to homotopy categories of abelian model categories.

Findings

New examples of stable categories with triangulated structures from abelian model categories

Description of pretriangulated structures of homotopy categories via stable categories

Extension of triangulated structure constructions to broader contexts

Abstract

Beligiannis and Marmaridis [\emph{Comm. in Algebra,} 22(12)(1994), 5021-5036] constructed the left and right triangulated structures on the stable categories of additive categories induced from some homological finite subcategories. We extend their results to slightly more general settings. As an application of our results we give some new examples of stable categories which have left or right triangulated structures from abelian model categories. An interesting outcome is that we can describe the pretriangulated structures of the homotopy categories of abelian model categories via the ones of stable categories.