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