Mal'cev, Protomodular, Homological and Semi-Abelian Categories

The book "Mal'cev, Protomodular, Homological and Semi-Abelian Categories" offers a comprehensive analysis of developments in algebra through the lens of category theory over the past fifteen years. It highlights the newly emerged and synthetic concepts of Mal'cev, protomodular, homological, and semi-abelian categories. These concepts draw attention to the fibration of points and enable a unified treatment of central algebraic topics such as homological lemmas, Noether isomorphisms, and commutator theory. By emphasizing examples and linking to universal algebra, the productivity of the fundamental categorical restriction is highlighted, allowing one to understand an object not from within through its elements, but from the outside through its relationships to the surrounding context. The book is aimed at researchers in category theory, homological theory, and universal algebra, and also serves as a textbook for graduate courses on these topics.