Homotopy lifting property

Summary

A map $p \colon E \to B$ has the homotopy lifting property with respect to a space $X$ if homotopies of maps $X \to B$ lift to $E$. The lift must start at a given lift of the starting map. This property defines the fibrations of homotopy theory: Hurewicz fibrations have it for all spaces, and Serre fibrations for all cubes. It is the formal dual of the homotopy extension property that defines cofibrations.