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This book gives a general outlook on homotopy theory; fundamental concepts, such as homotopy groups and spectral sequences, are developed from a few axioms and are thus available in a broad variety of contexts. Many examples and applications in topology and algebra are discussed, including an introduction to rational homotopy theory in terms of both differential Lie algebras and De Rham algebras. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given, including when the fundamental group is non-trivial. Moreover, the deep connection between the homotopy classification problems and the cohomology theory of small categories is demonstrated. The prerequisites of the book are few: elementary topology and algebra. Consequently, this account will be valuable for non-specialists and experts alike. It is an important supplement to the standard presentations of algebraic topology, homotopy theory, category theory and homological algebra.

Résumé. -On décrit les foncteurs polynomiaux, des groupes abéliens libres vers les groupes abéliens, comme des diagrammes de groupes abéliens dont on explicite les relations. Abstract (Polynomial functors and nonlinear Mackey functors)Polynomial functors from free abelian groups to abelian groups are described explicitely in the form of diagrams of abelian groups, that are maps between the crosseffects of the polynomial functor which satisfy a list of relations. The key is to use an appropriate notion of Mackey functor from the category of finite sets and surjections.Soient Ab la catégorie des groupes abéliens, et F (Z) sa sous-catégorie pleine dont les objets sont les groupes abéliens libres de type fini. On sait que la caté-gorie des foncteurs additifs de F (Z) vers Ab estéquivalenteà la catégorie Ab.

There are three basic constructions in literature which relate a 1-connected topological space X to a differential graded algebra. Adams and Hilton [2] constructed a chain algebra (with integer coefficients) for the loop space ~?X, a special version of which is Adams' cobar construction [1]. Later Quillen [13] associated a differential graded rational Lie algebra 2(X) to the space X, and Sullivan [14,15], using simplicial differential forms with rational coefficients, obtained a DG commutative cochain algebra for X. For these cochain algebras, Sullivan introduces the notion of minimal model, which corresponds to the Postnikov decomposition of a space.In this paper, we construct minimal models for chain algebras (over any field) and for rational DG Lie algebras. These minimal models correspond to the Eckmann-Hilton homology decomposition of a space. The algebraic construction of the minimal model uses algebraic versions of the Hurewicz and BlakersMassey theorems (2.6), (2.9). The corresponding minimal models for topological spaces are studied in § 3. § 0. Recollection of Notations and ResultsOur references for notations are [10] and [13]. For the reader's convenience, we here collect basic definitions and facts we shall use in this paper.