From Groups to Categorial Algebra: Introduction to Protomodular and Mal’tsev Categories (Compact Textbooks in Mathematics) by Dominique Bourn (PDF)

Description

This book gives a thorough and entirely self-contained, in-depth introduction to a specific approach to group theory, in a large sense of that word. The focus lie on the relationships which a group may have with other groups, via “universal properties”, a view on that group “from the outside”. This method of categorical algebra, is actually not limited to the study of groups alone, but applies equally well to other similar categories of algebraic objects.By introducing protomodular categories and Mal’tsev categories, which form a larger class, the structural properties of the category Gp of groups, show how they emerge from four very basic observations about the algebraic litteral calculus and how, studied for themselves at the conceptual categorical level, they lead to the main striking features of the category Gp of groups.Hardly any previous knowledge of category theory is assumed, and just a little experience with standard algebraic structures such as groups and monoids. Examples and exercises help understanding the basic definitions and results throughout the text.

User’s Reviews

Editorial Reviews: Review “The book is an excellent and well-thought introduction to the theory from its very first principles: apart from a limited knowledge of elementary abstract algebra, nothing is taken for granted in the introductory first pages. It is both an excellent and friendly reference for individual study and a well-organized reference text for a possible course of categorical algebra.” (Fosco Loregian, zbMATH 1403.18001, 2019) From the Back Cover This book gives a thorough and entirely self-contained, in-depth introduction to a specific approach to group theory, in a large sense of that word. The focus lie on the relationships which a group may have with other groups, via “universal properties”, a view on that group “from the outside”. This method of categorical algebra, is actually not limited to the study of groups alone, but applies equally well to other similar categories of algebraic objects.By introducing protomodular categories and Mal’tsev categories, which form a larger class, the structural properties of the category Gp of groups, show how they emerge from four very basic observations about the algebraic litteral calculus and how, studied for themselves at the conceptual categorical level, they lead to the main striking features of the category Gp of groups.Hardly any previous knowledge of category theory is assumed, and just a little experience with standard algebraic structures such as groups and monoids. Examples and exercises help understanding the basic definitions and results throughout the text. About the Author Dominique Bourn is Professor at the Université du Littoral in Calais, France. Read more

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