Editing Homotopy lifting property (section)

==Generalization: homotopy lifting extension property==
There is a common generalization of the homotopy lifting property and the [[homotopy extension property]].  Given a pair of spaces <math>X \supseteq Y</math>, for simplicity we denote <math>T \mathrel{:=} (X \times \{0\}) \cup (Y \times [0, 1]) \subseteq X\times [0, 1]</math>.  Given additionally a map <math>\pi \colon E \to B</math>, one says that ''<math>(X, Y, \pi)</math> has the '''homotopy lifting extension property''''' if:
* For any [[homotopy]] <math>f \colon X \times [0, 1] \to B</math>, and
* For any lifting <math>\tilde g \colon T \to E</math> of <math>g = f|_T</math>, there exists a homotopy <math>\tilde f \colon X \times [0, 1] \to E</math> which covers <math>f</math> (i.e., such that <math>\pi\tilde f = f</math>) and extends <math>\tilde g</math> (i.e., such that <math>\left.\tilde f\right|_T = \tilde g</math>).

The homotopy lifting property of <math>(X, \pi)</math> is obtained by taking <math>Y = \emptyset</math>, so that <math>T</math> above is simply <math>X \times \{0\}</math>.

The homotopy extension property of <math>(X, Y)</math> is obtained by taking <math>\pi</math> to be a constant map, so that <math>\pi</math> is irrelevant in that every map to ''E'' is trivially the lift of a constant map to the image point of <math>\pi</math>.