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Definition.

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

for each pair of map f:XEf:X\rightarrow E, and a homotopy H:X×IBH:X\times I\rightarrow B starting at H0=pfH_0=p\circ f, there exists a homotopy H~:X×IE\tilde{H}:X\times I\rightarrow E such that:

  • H~0=f\tilde{H}_0=f
  • pH~=Hp\tilde{H}=H

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

In categorical language, the following diagram commutes:

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