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.
Keywords: Model Category, Weak Factorization Systems, Two out of Three Property
2010 MSC: 18G55
Theory and Applications of Categories, Vol. 30, 2015, No. 36, pp 1163-1180.