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Mal’cev, protomodular, homological and semi-abelian categories. (English) Zbl 1061.18001
Mathematics and its Applications (Dordrecht) 566. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1961-0/hbk). xiii, 479 p. (2004).
The subject of non-abelian homological algebra has a curious history. Whilst it has been actively studied for over thirty years, it is only within the last decade that the definitive categorical setting for this study has emerged, largely through the work of the second author of this book. Although various authors, from the mid-1960s onwards, had made some progress in the search for a non-commutative version of the notion of abelian category, in which the diagram-chasing lemmas of homological algebra would become intrinsic properties of the category (as they do in abelian categories), it was still found necessary to use a great deal of ‘external’ machinery in order to study the homological properties of such categories. It was not until the 1990s that the second author of this book found the key to a truly intrinsic treatment of homology in the non-commutative case. The key turns out to lie in a distinctly unexpected place: namely the fibration of ‘pointed objects’ (that is, split epimorphisms in the underlying category). By analysing properties of this fibration, Bourn was led to introduce the central notion of protomodularity, around which all the other concepts studied in this book can be seen as clustering.
Protomodularity implies the ‘Mal’cev property’ that every reflexive relation is an equivalence relation; the latter property, named in honour of A. I. Mal’cev’s pioneering work on varieties of universal algebras possessing this property, had already been studied by several authors, taking up the challenge posed by J. D. H. Smith [in his book “Mal’cev Varieties”, Lect. Notes Math. 554 (1976; Zbl 0344.08002)] to provide an intrinsic categorical treatment of the ideas that he had developed for varieties of algebras. But protomodularity is significantly stronger; although the second author and G. Janelidze succeeded in finding a ‘Mal’cev-type condition’ characterizing the varieties of algebras which satisfy it, and although that condition turned out to have been studied previously in the context of universal algebra by A. Ursini, it had not attracted much attention in the latter context. It seems that the more sharply focused light of the categorical approach was needed in order to illuminate the true significance of this condition.
The aim of the book under review is to set out the material that has been developed around the concept of protomodularity over the last ten years, in the belief that it is now ripe for treatment in book form. The authors take a ‘step by step’ approach, devoting successive chapters to unital, Mal’cev, protomodular, homological and semi-abelian categories: this has a slightly repetitive effect, in that the same categories tend to come around each time as examples of the structure currently under discussion. As a result, the book is perhaps rather less useful as a reference work than it might have been (for example, a reader who wishes to know why the dual of an elementary topos is protomodular will not find a self-contained proof here, but rather one which refers back to the proof in the previous chapter that such categories are Mal’cev), but the repetition will doubtless be beneficial for students using the book as a textbook. And it should make a first-rate textbook: the authors’ choice of material is judicious (with a few minor exceptions, for which see below), and the style of the book is clear and easy to follow (though the fact that the first language of both authors is French is deducible from almost every page).
Since the book is likely to remain the definitive reference for this material, it is slightly unfortunate that there are one or two minor errors and omissions. The most serious of the former occurs in Example 3.1.11: the ‘proof’ of protomodularity of the variety of Heyting algebras, allegedly quoted from a recent paper of the reviewer, is in fact misquoted, and it is not a valid proof. Among the omissions are the real point of the reviewer’s paper just mentioned (namely, that Heyting algebras form an example of a variety where it is necessary to use more than two operations to verify the Bourn-Janelidze condition for protomodularity) and, more bizarrely, the reviewer’s name where this paper is listed in the bibliography. Another unfortunate omission is that of the relationship between the authors’ notion of ‘semi-associativity’ for a Mal’cev operation and that of ‘weak associativity’ introduced by the reviewer and M. C. Pedicchio: although this is hinted at by a throwaway remark at the top of page 334, it would have taken hardly any more space to establish the actual relationship (namely, that a semi-associative operation is weakly associative, and although the converse is false, a variety with a weakly associative Mal’cev operation and at least one constant also has a semi-associative operation). Again, although the (easily established) fact that a non-degenerate protomodular variety must have at least one constant is implicit in the proof of Theorem 3.1.6, it is never explicitly stated.
It would be pleasant (and well-deserved) if the book succeeded in selling well enough to justify a second edition, in which these minor defects could be remedied.

MSC:
18-02 Research exposition (monographs, survey articles) pertaining to category theory
08B05 Equational logic, Mal’tsev conditions
18B99 Special categories
18C10 Theories (e.g., algebraic theories), structure, and semantics
18D30 Fibered categories
18E10 Abelian categories, Grothendieck categories
18G50 Nonabelian homological algebra (category-theoretic aspects)
20J05 Homological methods in group theory
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