BUILDING A MODEL CATEGORY OUT OF COFIBRATIONS AND FIBRATIONS: THE TWO OUT OF THREE PROPERTY FOR WEAK EQUIVALENCES

The purpose of this note is to understand the two out of three property of the model category in terms of the weak factorization systems. We will show that if a category with classes of trivial cofibrations, cofibrations, trivial fibrations, and fibrations is given a simplicial structure similar to that of the simplicial model category, then the full subcategory of cofibrant and fibrant objects has the two out of three property, and we will give a list of necessary and sufficient conditions in terms of the simplicial structure for the associated canonical "weak equivalence class" to have the two out of three property.