Minimal fibrations and the organizing theorem of simplicial homotopy theory

TL;DR

This paper provides an elementary proof of Quillen's organizing theorem for simplicial homotopy theory, avoiding complex classification and topological methods, thereby simplifying the foundational understanding of the subject.

Contribution

It introduces a new, elementary proof of the organizing theorem that does not rely on principal bundle classification or topological techniques.

Findings

Simplifies the proof of the organizing theorem

Eliminates the need for principal bundle classification

Provides a more accessible foundation for simplicial homotopy theory

Abstract

Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory of principal bundles. Thus, Goerss-Jardine appealed to topological methods for the verification. In this paper we give a new proof of this organizing theorem of simplicial homotopy theory which is elementary in the sense that it does not use the classifying theory of principal bundles or appeal to topological methods.