The book exposes the theory of protomodular and Mal’tsev categories, presenting these two topics from their very first principles. The theory of protomodular categories arises in order to axiomatize the behavior of algebraic categories such as Grp – the category of groups – which, although lacking familiar properties such as those of abelian categories, are particularly well-behaved and a paradigmatic example of algebraic categories.
A “protomodular category” is defined by a list of axioms abstracting the behavior of the group category: this terse axiomatization yields all sorts of properties mimicking the behaviour of Grp, the monics are all and the only morphisms with zero kernel, all the epics are the cokernel of their kernel, there is a class of distinct mono called ‘normal’ (which mimicking the inclusions \( H \vartriangleleft G \) of normal subgroups), and internal equivalence relations are particularly well-behaved. This last property is what defines Mal’tsev categories, which are subsequently introduced and studied. Then the author moves on to study exactness and linearity properties (these latter are of course of interest for the extension of homological algebra to these contexts, see e.g. the possibility to define various “animal lemmas” in an exact protomodular category).
Notably, this leads to the theory of semi-abelian categories: pointed, Barr-exact, protomodular categories with finite co-products. In a semi-abelian category it is possible to recast the results of homological algebra in a genuinely non-additive context. Indeed, the semi-abelian framework is perfectly suited to the study of non-abelian (co) homology and the corresponding homotopy theory.
Although this aspect of the theory is not investigated, a large number of pointer to the existing literature is given, and the reader who addresses the reasoned study of the volume is perfectly equipped to address the state of the art in the subject.
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.