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Full Description
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).
Contents
Dictionary.- I. Categories of Fractions.- 1. Categories of Fractions. Categories of Fractions and Adjoint Functors.- 2. The Calculus of Fractions.- 3. Calculus of Left Fractions and Direct Limits.- 4. Return to Paragraph 1.- II. Simplicial Sets.- 1. Functor Categories.- 2. Definition of Simplicial Sets.- 3. Skeleton of a Simplicial Set.- 4. Simplicial Sets and Category of Categories.- 5. Ordered Sets and Simplicial Sets. Shuffles.- 6. Groupoids.- 7. Groupoids and Simplicial Sets.- III. Geometric Realization of Simplicial Sets.- 1. Geometric Realization of a Simplicial Set.- 4. Kelley Spaces.- 3. Exactness Properties of the Geometric Realization Functor.- 4. Geometric Realization of a Locally Trivial Morphism.- IV. The Homotopic Category.- 1. Homotopies.- 2. Anodyne Extensions.- 3. Kan Complexes.- 4. Pointed Complexes.- 5. Poincaré Group of a Pointed Complex.- V. Exact Sequences of Algebraic Topology.- 1. 2-Categories.- 2. Exact Sequences of Pointed Groupoids.- 3. Spaces of Loops.- 4. Exact Sequences: Statement of the Theorem and Invariance.- 5. Proof of Theorem 4.2.- 6. Duality.- 7. First Example: Pointed Topological Spaces.- 8. Second Example: Differential Complexes of an Abelian Category.- VI. Exact Sequences of the Homotopic Category.- 1. Spaces of Loops.- 2. Cones.- 3. Homotopy Groups.- 4. Generalities on Fibrations.- 5. Minimal Fibrations.- VII. Combinatorial Description of Topological Spaces.- 1. Geometric Realization of the Homotopic Category.- 2. Geometric Realization of the Pointed Homotopic Category.- 3. Proof of Milnor's Theorem.- Appendix I. Coverings.- 1. Coverings of a Groupoid.- 2. Coverings of Groupoids and Simplicial Coverings.- 3. Simplicial Coverings and Topological Coverings.- Appendix II. The Homology Groups of a Simplicial Set.- 2. The ReducedHomology Group of a Pointed Simplicial Set.- 3. The Spectral Sequence of Direct Limits.- 4. The Spectral Sequence of a Fibration.- Index of Notations.- Terminological Index.
