Another approach to the Kan-Quillen model structure

TL;DR

This paper presents a new combinatorial proof that the comparison map from a simplicial set to its ex-infinity image is a weak homotopy equivalence, leading to a novel construction of the Kan-Quillen model structure without topological spaces.

Contribution

It introduces a combinatorial approach to the Kan-Quillen model structure, avoiding the use of topological spaces and minimal fibrations.

Findings

Comparison map is a weak homotopy equivalence

Presented as a strong anodyne extension

Provides a new combinatorial construction of the model structure

Abstract

By careful analysis of the comparison map from a simplicial set to its image under Kan's ex-infinity 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 are able to quickly deduce some basic facts about ex-infinity and hence provide a new construction of the Kan-Quillen model structure on simplicial sets, which avoids the use of topological spaces or minimal fibrations.