[Paper Review] Extriangulated categories, Hovey twin cotorsion pairs and model structures

Hiroyuki Nakaoka, Yann Palu|arXiv (Cornell University)|Jan 1, 2019

Algebraic structures and combinatorial modelsMathematics4 references115 citations

TL;DR

The paper introduces extriangulated categories as a simultaneous framework for exact and triangulated categories, establishes a bijective correspondence between Hovey twin cotorsion pairs and admissible model structures, and develops a localization-homotopy framework to study reduction and mutation of cotorsion pairs.

ABSTRACT

We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated structure. This gives a natural framework to formulate reduction and mutation of cotorsion pairs, applicable to both exact categories and triangulated categories. These results can be thought of as arguments towards the view that extriangulated categories are a convenient setup for writing down proofs which apply to both exact categories and (extension-closed subcategories of) triangulated categories.

Motivation & Objective

• Generalize concepts from exact and triangulated categories using an extriangulated framework.
• Study cotorsion pairs in this unified setting and relate them to model structures.
• Develop a correspondence between Hovey twin cotorsion pairs and admissible model structures.
• Describe how the homotopy category can be triangulated via certain quotients and localizations.
• Provide a framework for reduction and mutation of cotorsion pairs applicable to both exact and triangulated settings.

Proposed method

• Define and employ the E-extension framework to capture Ext^1-like data in an additive category.
• Introduce an additive realization of E-extensions to form extriangulated categories (ET1–ET4).
• Develop cotorsion pairs in extriangulated categories and study associated adjoint functors.
• Establish a bijective correspondence between Hovey twin cotorsion pairs and admissible model structures.
• Describe the triangulation of the homotopy category via localization and ideal quotients.

Experimental results

Research questions

• RQ1 How can cotorsion pairs be formulated in the extriangulated setting to generalize exact and triangulated cases?
• RQ2 What is the precise relationship between Hovey twin cotorsion pairs and admissible model structures in extriangulated categories?
• RQ3 How can one realize a triangulated structure on the homotopy category arising from model structures in this framework?
• RQ4 In what way can reduction and mutation of cotorsion pairs be formulated uniformly for exact and triangulated categories through extriangulated categories?

Key findings

• There is a bijective correspondence between certain pairs of cotorsion pairs (Hovey twin cotorsion pairs) and admissible model structures.
• The homotopy category can be realized as an ideal quotient and given a triangulated structure.
• The extriangulated framework unifies reduction and mutation of cotorsion pairs across exact and triangulated settings.
• The results provide a unified proof strategy showing extriangulated categories accommodate proofs applicable to both exact and triangulated contexts.