  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.
