Definition.

A map $p:E\rightarrow B$ is said to have the homotopy lifting property ( HLP ) with respect to a space $X$, if:

for each pair of map $f:X\rightarrow E$, and a homotopy $H:X\times I\rightarrow B$ starting at $H_0=p\circ f$, there exists a homotopy $\tilde{H}:X\times I\rightarrow E$ such that:

• $\tilde{H}_0=f$
• $p\tilde{H}=H$

The map $p$ is said to be a (Hurewicz) fibration if it has HLP with respect to all spaces.

In categorical language, the following diagram commutes: