Simplicial sets are discrete analogs of topological spaces. They have
played a central role in algebraic topology ever since their
introduction in the late 1940s, and they also play an important role in
other areas such as geometric topology and algebraic geometry. On a
formal level, the homotopy theory of simplicial sets is equivalent to
the homotopy theory of topological spaces. In view of this equivalence,
one can apply discrete, algebraic techniques to perform basic
topological constructions. These techniques are particularly appropriate
in the theory of localization and completion of topological spaces,
which was developed in the early 1970s.
Since it was first published in 1967, Simplicial Objects in Algebraic
Topology has been the standard reference for the theory of simplicial
sets and their relationship to the homotopy theory of topological
spaces. J. Peter May gives a lucid account of the basic homotopy theory
of simplicial sets, together with the equivalence of homotopy theories
alluded to above. The central theme is the simplicial approach to the
theory of fibrations and bundles, and especially the algebraization of
fibration and bundle theory in terms of twisted Cartesian products. The
Serre spectral sequence is described in terms of this algebraization.
Other topics treated in detail include Eilenberg-MacLane complexes,
Postnikov systems, simplicial groups, classifying complexes, simplicial
Abelian groups, and acyclic models.
Simplicial Objects in Algebraic Topology presents much of the
elementary material of algebraic topology from the semi-simplicial
viewpoint. It should prove very valuable to anyone wishing to learn
semi-simplicial topology. [May] has included detailed proofs, and he
has succeeded very well in the task of organizing a large body of
previously scattered material.--Mathematical Review
