This entry is about the monograph
(Initially it will be in the form of a book review. The initial version is an edited and adapted version of a review of the book by Tim Porter that appeared in Proceedings of the Edinburgh Mathematical Society (Series 2) (2002), 45 : 761-762. When the entry has been influenced by other sources, please remove this box)
The developments that lead to this book are due to some deeply based analogies between some initially very different looking mathematical theories, one in algebra, another in topology.
Galois originally developed some of elements of what was to become Galois theory in an attempt to understand polynomial equations, continuing work of Abel and others. In modern language, working over a base field, $K$, a field extension $K\subset L$ is a Galois extension when every element of $L$ is the root of a polynomial in $K[X]$, which factors in $L[X]$ into linear factors and all of whose roots are simple. The group $Gal[L:K]$ of this extension is the group of field automorphisms of $L$ which fix all elements of $K$ and the classical Galois theorem asserts that when $L$, considered as a $K$-vector space, is finite dimensional, the subgroups $G\subseteq Gal[L:K]$ of the Galois group classify the intermediate field extensions $K\subseteq M\subseteq L$. The themes to note in this classical theory are (i) splitting into simpler structures, (ii) groups of automorphisms and (iii) intermediate structures classified by subgroups.
The second theory is that of covering spaces in topology. A covering map $\alpha : X\to B$ is one that has the property that every point of $B$ has an open neighbourhood whose inverse image by $\alpha$ is a disjoint union of open subsets each of which is mapped homeomorphically onto it by $\alpha$. Given any reasonably ‘locally nice’ space, there is a universal connected covering space, $p : \widetilde{B}\to B$, such that all connected covering spaces of $B$ are quotients of $\widetilde{B}$. Classification of the connected coverings is by subgroups of the automorphism group of $p$. Of course, this automorphism group is isomorphic to the (Poincaré) fundamental group of $B$ (under suitable local conditions).
This topological theory of covering spaces has some similarities to Galois theory. Again one has automorphism groups and a correspondence between intermediate structures (this time quotients not subobjects) and perhaps some notion of splitting - there is an open cover of $B$ over each part of which the covering splits up as a family of isomorphic pieces.
These two theories, thus, do look vaguely similar, but automorphism groups are very common in mathematics and even that sort of ‘Galois’ correspondence is not that uncommon, so surely any similarities must not be due to anything really deep! The story of how the deep connections between them became apparent is quite long and here is not the place to explore it in any detail. It involves function spaces and Riemann surfaces as well as a lot else that is central to modern pure maths (if you want a good source for the theory see the beautiful book (see below) by Douady and Douady.
Borceux and Janleidze’s book, Galois theories, traces a greatly extended mathematical path beyond that semi-classical link between Galois theory and coverings, but starts at a fairly elementary beginning. It describes classical Galois theory, then turns to its extension to infinitary field extensions, to étale algebras that became the foundation for the work of Grothendieck on the fundamental group of schemes (SGA1). This was at the same time Galois theory and covering space theory, although for spaces for which there was no question of being able to define a fundamental group using paths. (This idea was fundamental in Grothendieck’s later attempts to develop a higher dimensional version of this Poincaré-Galois theory in his manuscript, Pursuing Stacks.)
This book explores the connections between these theories, searching for further cases of the general ‘scenario’ and tries to strip back the superficial structure to reveal aspects of what are the essential features of all these theories.
The book assumes a certain knowledge of algebra and general topology, together with some familiarity with the elementary language of category theory (categories, functors, natural transformations, limits and adjoint functors). Its starts with a short trip through the theory of field extensions, then goes on to look at Grothendieck’s extension of this to algebras.
Chapter 3 handles infinitary Galois theory. Here profinite spaces and profinite groups are introduced. They are also useful in the following chapter where the extension of Galois theory to commutative rings (due to Magid, see reference below) is treated. The Pierce spectrum and Stone duality are handled clearly and simply laying the base for Magid’s profinite Galois groupoid.
With that first layer of categorical abstraction (Janelidze’s abstract categorical Galois theory) in place, other applications can be explored. Given the introduction to the area in this review, it may seem strange that covering maps are only introduced in Chapter 6, but here they can be very neatly described and handled, beautifully illustrating and enriching the earlier abstraction.
The final chapter describes one of the most important recent advances in topos theory, giving an introduction to the Joyal–Tierney classification of Grothendieck toposes as sheaves on localic groupoids. The book ends with a look at other directions the theory has taken beyond those handled in detail. This section is particularly valuable as it should set the scene for future research.
It introduces many deep important concepts of algebra and category theory, introduces, motivates and uses in an interesting way. As one would expect, this means that the later chapters are sometimes much harder going than the earlier ones, but the writing and structure of the book is such that the transition is fairly gradual.
Some researchers will probably feel that some of the exercises should have been worked out in detail, but given the starting assumptions of the authors, and the reasonably good set of references, there were bound to be such omissions.
$n$lab entries categorical Galois theory, Grothendieck's Galois theory
Grothendieck et al., SGA1
R and A. Douady, Algèbre et théories galoisiennes, Fernand Nathan, 1977.
A.R.Magid, The separable Galois theory of commutative rings, Marcel Dekker, 1974.