Fibration and cofibration

Summary

In a model category, the fibrations and the cofibrations are two of the three distinguished classes of morphisms, besides the weak equivalences. They abstract the Serre fibrations and the cofibrations of topology: maps along which homotopies lift, and inclusions along which homotopies extend. Each class is determined by the other together with the weak equivalences, through lifting properties. Fibrant and cofibrant objects are the good objects on which homotopy classes of maps compute the homotopy category.