Mal'cev, Protomodular, Homological and Semi-Abelian Categories / Edition 1

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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.

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Editorial Reviews

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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)

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Product Details

  • ISBN-13: 9781402019616
  • Publisher: Springer Netherlands
  • Publication date: 2/29/2004
  • Series: Mathematics and Its Applications (closed) Series, #566
  • Edition description: 2004
  • Edition number: 1
  • Pages: 480
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.06 (d)

Table of Contents

Metatheorems 1
1 Intrinsic centrality 11
2 Mal'cev categories 125
3 Protomodular categories 229
4 Homological categories 273
5 Semi-abelian categories 319
6 Strongly protomodular categories 345
7 Essentially affine categories 371
App. A.1 Algebraic theories 399
App. A.2 Internal relations 411
App. A.3 Internal groupoids 417
App. A.4 Variations on epimorphisms 427
App. A.5 Regular and exact categories 436
App. A.6 Monads 449
App. A.7 Fibrations 461
Classification table of the fibration of points 466
Bibliography 467
Index of symbols 473
Index of definitions 475
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