Priles of one-sided triangulated categories and exact model categories

TL;DR

This paper introduces priles of one-sided triangulated categories, providing a framework to recover the pretriangulated structure of homotopy categories in exact model categories, bridging concepts in triangulated and model category theory.

Contribution

It defines the concept of priles of one-sided triangulated categories and applies it to recover structures in exact model categories, advancing the understanding of their homotopy categories.

Findings

Priles unify two one-sided triangulated categories via a common subcategory.

The framework recovers the pretriangulated structure of homotopy categories.

Application to exact model categories demonstrates the framework's utility.

Abstract

We introduce the notion of a prile of one-sided triangulated categories. Roughly speaking, a prile consists of two one-sided triangulated categories having a common full subcategory which inherits a pretriangulated structure from these ambient categories. The main example arises from exact model categories. This allows us to recover the pretriangulated structure of the homotoy category of an exact model category.