Simplicial Set - Homotopy Theory of Simplicial Sets

Homotopy Theory of Simplicial Sets

In the category of simplicial sets one can define fibrations to be Kan fibrations. A map of simplicial sets is defined to be a weak equivalence if the geometric realization is a weak 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 satisfies the axioms for a proper closed simplicial model category.

A key turning point of the theory is that the 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 abstract nonsense. Furthermore, the geometric realization and singular functors give a Quillen equivalence of closed model categories inducing an equivalence of homotopy categories

|•|: Ho(S) ↔ Ho(Top) : S

between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of 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).