Preface The subject of topology can be described as the study of the category Top of all topological spaces and the continuous maps between them. But many topological problems, and their solutions, do not change if the maps involved are replaced with ‘continuous deformations’ of themselves. The equivalence relation—called homotopy—generated by continuous deformations of maps respects composition, so that there is a ‘quotient’ homotopy category hTop and a functor Top hTop. Homotopy theory is the study of this functor. Thus homotopy theory is not entirely confined to the category hTop: it is frequently necessary, or at least useful, to use constructions available only in Top in order to prove statements that are entirely internal to hTop and the homotopy category hTop can shed light even on questions in Top that are not homotopy invariant. History. The core of the subject I’m calling ‘classical homotopy theory’ is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This includes the notions of fibration and cofibration, CW complexes, long fiber and cofiber sequences, loop space, suspension, and so on. Brown’s representability theorems show that homology and cohomology are also contained in classical homotopy theory. One of the main complications in homotopy theory is that many, if not most, diagrams in the category hTop do not have limits or colimits. Thus many theorems were proved using occasionally ingenious and generally ad hoc constructions performed in the category Top. Eventually many of these constructions were codified in the dual concepts of homotopy colimit and xvii
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