Editing Simplicial set (section)

==Homotopy theory of simplicial sets==
In order to define a [[Model category|model structure]] on the category of simplicial sets, one has to define fibrations, cofibrations and weak equivalences. One can define [[Fibration of simplicial sets|fibrations]] to be [[Kan fibration]]s.  A map of simplicial sets is defined to be a weak equivalence if its geometric realization is a [[Weak homotopy equivalence|weak homotopy equivalence of spaces]].  A map of simplicial sets is defined to be a [[cofibration]] if it is a [[monomorphism]] of simplicial sets.  It is a difficult theorem of [[Daniel Quillen]] that the category of simplicial sets with these classes of morphisms becomes a model category, and indeed satisfies the axioms for a [[proper model category|proper]] [[closed model category|closed]] [[simplicial model category]].

A key turning point of the theory is that the geometric realization of a Kan fibration is a [[Serre fibration]] of spaces.  With the model structure in place, a homotopy theory of simplicial sets can be developed using standard [[homotopical algebra]] methods.  Furthermore, the geometric realization and singular functors give a [[Quillen adjunction|Quillen equivalence]] of [[closed model category|closed model categories]] inducing an equivalence

:|•|: ''Ho''('''sSet''') ↔ ''Ho''('''Top''')

between the [[homotopy category]] for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of continuous maps between them. It is part of the general
definition of a Quillen adjunction that the right adjoint functor (in this case, the singular set functor) carries fibrations (resp. trivial fibrations) to fibrations (resp. trivial fibrations).