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

ISBN-13: 9781402019616 / Angielski / Twarda / 2004 / 480 str.

Dominique Bourn; Francis Borceux

Opis

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.

Recenzje

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)

Spis treści

Preface
Metatheorems
0.1 The Yoneda embedding
0.2 Pointed categories

1 Intrinsic centrality
1.1 Spans and relations
1.2 Unital categories
1.3 Cooperating and central morphisms
1.4 Commutative objects
1.5 Symmetrizable morphisrns
1.6 Regular unital categories
1.7 Associated abelian object
1.8 Strongly unital categories
1.9 Gregarious objects
1.10 Linear and additive categories
1.11 Antilinear and antiadditive categories
1.12 Complemented subobjects

2 Mal’cev categories
2.1 Slices, coslices and points
2.2 Mal’cev categories
2.3 Abelian objects in Mal’cev categories
2.4 Naturally Mal’cev categories
2.5 Regular Mal’cev categories
2.6 Connectors in Mal’cev categories
2.7 Connector and cooperator
2.8 Associated abelian object and commutator
2.9 Protoarithmetical categories
2.10 Antilinear Mal’cev categories
2.11 Abelian groupoids

3 Protomodular categories
3.1 Definition and examples
3.2 Normal subobjects
3.3 Couniversal property of the product
3.4 Groupoids, protomodularity and normality

4 Homological categories
4.1 The short five lemma
4.2 The nine lemma
4.3 The Noether isomorphism theorems
4.4 The snake lemma
4.5 The long exact homology sequence
4.6 Examples of homological categories

5 Semi-abelian categories
5.1 Definition and examples
5.2 Semi-direct products
5.3 Semi-associative Mal’cev varieties

6 Strongly protomodular categories
6.1 Centrality and normality
6.2 Normal subobjects in the fibres
6.3 Normal functors
6.4 Strongly protomodular categories
6.5 A counterexample
6.6 Connector and cooperator

7 Essentially affine categories
7.1 The fibration of points
7.2 Essentially affine categories
7.3 Abelian extensions

Appendix
A.1 Algebraic theories
A.2 Internal relations
A.3 Internal groupoids
A.4 Variations on epimorphisms
A.5 Regular and exact categories
A.6 Monads
A.7 Fibrations

Classification table of the fibration of points
Bibliography
Index of symbols
Index of definitions

O autorze

Francis Borceux is Professor of mathematics at the University of Louvain since many years. He has developed research in algebra and essentially taught geometry, number theory and algebra courses and he has been dean of the Faculty of Sciences of his University and chairman of the Mathematical Committee of the Belgian National Scientific Research Foundation.