Brown's first five chapters are true topological background material,
stretching as they do from the topology of the real line to a discussion
of projective ("and other") spaces. It is only in the sixth chapter (p.
201 ff.) that we get to the all-important fundamental groupoid, but
thereafter things get off the ground very swiftly: homotopy theory,
cofibrations, computing fundamental groupoids (Van Kampen, the Jordan
Curve Theorem revisited), covering spaces, orbit spaces, and, indeed,
orbit groupoids. A broad palette.
I do believe in the general efficacy of the general categorical approach
in mathematics ...., and I find Brown's philosophy both attractive and
convincing. To wit (p. xx):
"In mathematics, and in many areas, analogies are not between objects
themselves, but between the relations between these objects. We will
define many constructions by their relations to all other objects of the
same type - this is called a 'universal property' ... All this is the
essence of the 'categorical approach', ... a major unifying force in the
mathematics of the twentieth century."
Two final observations. The back cover of Topology and Groupoids
displays a Venn diagram suggesting that, to borrow another word from
Grothendieck, the yoga of groupoids should be amenable eventually to
include, or engulf, such objects as groups, group actions, bundles of
groups (!), and even sets and equivalence relations. This in itself is a
very exciting prospect, alone worth the price of admission. ....... The
book is well written, indeed it is really a monograph composed by an
insider and an expert; it is very serious mathematics presented in a
sound pedagogical style: it is a very readable book equipped with fine
examples and many exercises; and its impact should be felt beyond the
confines of topology, even as topologists should be attracted to this
material most strongly.
Topology and Groupoids is an impressive work which should be given a
Review by Michael Berg for the Mathematical Association of America
Studied topology with JHC Whitehead and MG Barratt. Lectured at Liverpool, Hull and Professor at Bangor since 1970. Over 170 publications, (124 on MathSciNet, with 49 co-authors) on topology, algebra, category theory, and 36 on popularisation and teaching. Originator of the terms "convenient category" and "higher dimensional algebra". His latest publication is R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011) which gives an exposition of aspects of 40 years of research on developing applications of higher groupoids in topology.