Stable Infinity Categories

TL;DR

This paper provides an expository overview of stable infinity categories, proving their homotopy categories are triangulated and detailing their constructions and universal properties.

Contribution

It offers a comprehensive explanation of stable infinity categories, including their properties, constructions, and relation to derived categories, with new proofs and characterizations.

Findings

Homotopy category of a stable infinity category is triangulated

Stable infinity categories are closed under various constructions

Derived categories can be realized as homotopy categories of stable infinity categories

Abstract

This paper is an expository account of the theory of stable infinity categories. We prove that the homotopy category of a stable infinity category is triangulated, and that the collection of stable infinity categories is closed under a variety of constructions. We also explain how to construct the derived category of an abelian category (with enough projective objects) as the homotopy category of a suitable stable infinity category; moreover, we characterize this stable infinity category by a universal mapping property.