Introduction to extriangulated categories

Extriangulated categories have been introduced by Nakaoka and Palu to generalize two important families of categories: exact and triangulated categories. The definition of extriangulated categories let us prove analogue theorems in a unified framework. The first chapter introduces exact categories, a generalization of abelian categories. After the proof of the main theorems, the focus shifts on generalization of concept of abelian categories in the exact category theory. An important family of additive categories are idempotent complete and weakly idempotent complete categories which possess useful properties when studied as exact categories. The subsequent sections regards acyclic complexes, a generalization of exact complexes of abelian categories. The second section has as main focus K(A) and some properties that are central to the definition of triangulated categories. A fundamental operation in category theory is localization which has a rich theory in triangulated framework. Another important example of localization are Serre quotient, obtained by localizing abelian categories by subcategories closed under extensions. The last section regards extriangulated categories, which generalize exact and triangulated categories, and the links with the latter two theories. The main focus of the last section is localization of extriangulated categories which unifies localization of triangulated categories and Serre quotients.