Calculus of Fractions and Homotopy Theory - ISBN:0387037772

Calculus of Fractions and Homotopy Theory

Date: 19 June 2011, 05:37

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The main purpose of the present work is to present to the reader a particularly nice category for the study of homotopy, namely the homotopic category. 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 to a category of fractions of the category of topological spaces modulo homotopy, and to the category of Kan complexes modulo homotopy.
In order to define our homotopic category, it appears useful to follow as closely as possible methods which have proved efficacious in homological algebra. Our category is thus the "topological" analogue of the derived category of an abelian category.

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