Another approach to the Kan\u2013Quillen model structure

Abstract

By careful analysis of the embedding of a simplicial set into its image under Kan’s mathop {mathop {mathsf {Ex}}^infty } functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about mathop {mathop {mathsf {Ex}}^infty } and hence provide a new construction of the Kan–Quillen model structure on simplicial sets, one which avoids the use of topological spaces or minimal fibrations.