The origin of simplicial homotopy theory coincides with the beginning of algebraic topology almost a century ago. The thread of ideas started with the work
of Poincar´e and continued to the middle part of the 20th century in the form
of combinatorial topology. The modern period began with the introduction of
the notion of complete semi-simplicial complex, or simplicial set, by EilenbergZilber in 1950, and evolved into a full blown homotopy theory in the work of
Kan, beginning in the 1950s, and later Quillen in the 1960s.
The theory has always been one of simplices and their incidence relations,
along with methods for constructing maps and homotopies of maps within these
constraints. As such, the methods and ideas are algebraic and combinatorial and,
despite the deep connection with the homotopy theory of topological spaces, exist completely outside any topological context. This point of view was effectively
introduced by Kan, and later encoded by Quillen in the notion of a closed model
category. Simplicial homotopy theory, and more generally the homotopy theories
associated to closed model categories, can then be interpreted as a purely algebraic enterprise, which has had substantial applications throughout homological
algebra, algebraic geometry, number theory and algebraic K-theory. The point
is that homotopy is more than the standard variational principle from topology
and analysis: homotopy theories are everywhere, along with functorial methods
of relating them.
This book is, however, not quite so cosmological in scope. The theory has
broad applications in many areas, but it has always been quite a sharp tool
within ordinary homotopy theory — it is one of the fundamental sources of
positive, qualitative and structural theorems in algebraic topology. We have
concentrated on giving a modern account of the basic theory here, in a form
that could serve as a model for corresponding results in other areas.
This book is intended to fill an obvious and expanding gap in the literature.
The last major expository pieces in this area, namely [33], [67], [61] and [18],
are all more than twenty-five years old. Furthermore, none of them take into
account Quillen’s ideas about closed model structures, which are now part of the
foundations of the subject.