We then showed that, in a model category, a map i : A --> B is a cofibration if and only if it has the left lifting property (LLP) with respect to every map p that is both a fibration and a weak equivalence. We used this to show that every isomorphism is a cofibration and that the co-base-change of a cofibration is a cofibration. Similar statements hold for each of the following three classes of maps: maps that are both cofibrations and weak equivalences, fibrations, and maps that are both fibrations and weak equivalences.
Finally, we introduced the notion of a cylinder object, a path object, left homotopy, and right homotopy. We shall show next time that if X is cofibrant and if Y is fibrant then, on the set Hom(X,Y) of maps from X to Y, the relations of left homotopy and right homotopy agree and are equivalence relations.