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
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
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