A Note on “The Homotopy Category is a Homotopy Category”

In his paper with the title, “The Homotopy Category is a Homotopy Category”, Arne Strøm shows that the category Top of topo- logical spaces satisfies the axioms of an abstract homotopy category in the sense of Quillen. In this study, we show by examples that Quillen’s model structure on Top fails to capture some of the subtleties of classical homotopy theory and also, we show that the whole of classical homo-topy theory cannot be retrieved from the axiomatic approach of Quillen. Thus, we show that model category is an incomplete model of classical homotopy theory.