© 2004

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


Part of the Mathematics and Its Applications book series (MAIA, volume 566)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Francis Borceux, Dominique Bourn
    Pages 1-9
  3. Francis Borceux, Dominique Bourn
    Pages 11-123
  4. Francis Borceux, Dominique Bourn
    Pages 125-228
  5. Francis Borceux, Dominique Bourn
    Pages 229-271
  6. Francis Borceux, Dominique Bourn
    Pages 273-317
  7. Francis Borceux, Dominique Bourn
    Pages 319-343
  8. Francis Borceux, Dominique Bourn
    Pages 345-370
  9. Francis Borceux, Dominique Bourn
    Pages 371-398
  10. Back Matter
    Pages 399-479

About this book


The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory.
The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment.
The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.


Abelian group Category theory algebra fibrations homology

Authors and affiliations

  1. 1.Université de LouvainLouvain-la-NeuveBelgium
  2. 2.Université du LittoralCalaisFrance

Bibliographic information


From the reviews:

"This monograph gives, from a categorical point of view, a coherent unified presentation of several aspects of universal algebra. … This is done in an elegant, unifying way via introducing several new concepts concerning objects, morphisms, and categories. … It contains up-to-date results and it is mainly addressed to working researchers in the field … . The exposition is well organized and the presentation is smooth. … In conclusion the monograph fills a gap in the literature and certainly it will be a standard reference in the future." (Apostolos D. Beligiannis, Mathematical Reviews, 2005e)

"The aim of the book under review is to set out the material that has been developed around the concept of protomodularity … . The authors take a ‘step by step’ approach … . As a result, the book is … beneficial for students using the book as a textbook. And it should make a first rate textbook: the authors’ choice of materials is judicious … and the style of the book is clear and easy to follow." (Peter T. Johnstone, Zentralblatt MATH, Vol. 1061 (12), 2005)