The main purpose of the present work is to present to the reader a
particularly nice category for the study of homotopy, namely the homo-
topic category (IV). This category is, in fact, - according to Chapter
VII and a well-known theorem of J. H. C. WHITEHEAD - equivalent to the
category of CW-complexes modulo homotopy, i.e. the category whose
objects are spaces of the homotopy type of a CW-complex and whose
morphisms are homotopy classes of continuous mappings between such
spaces. It is also equivalent (I, 1.3) to a category of fractions of the
category of topological spaces modulo homotopy, and to the category of
Kan complexes modulo homotopy (IV). In order to define our homotopic
category, it appears useful to follow as closely as possible methods
which have proved efficacious in homo- logical algebra. Our category is
thus the" topological" analogue of the derived category of an abelian
category (VERDIER). The algebraic machinery upon which this work is
essentially based includes the usual grounding in category theory -
summarized in the Dictionary - and the theory of categories of fractions
which forms the subject of the first chapter of the book. The merely
topological machinery reduces to a few properties of Kelley spaces
(Chapters I and III). The starting point of our study is the category,10
Iff of simplicial sets (C.S.S. complexes or semi-simplicial sets in a
former terminology).