Editing Homotopy (section)

=== Lifting and extension properties ===
{{main|Homotopy lifting property}}
If we have a homotopy {{nowrap|1=''H'' : ''X'' &times; [0,1] &rarr; ''Y''}} and a cover {{nowrap|1=''p'' : <span style="text-decoration: overline;">''Y''</span> &rarr; ''Y''}} and we are given a map {{nowrap|1=<span style="text-decoration: overline;">''h''</span><sub>0</sub> : ''X'' &rarr; <span style="text-decoration: overline;">''Y''</span>}} such that {{nowrap|1=''H''<sub>0</sub> = ''p'' ○ <span style="text-decoration: overline;">''h''</span><sub>0</sub>}}   (<span style="text-decoration: overline;">''h''</span><sub>0</sub> is called a [[Lift (mathematics)|lift]] of ''h''<sub>0</sub>), then we can lift all ''H'' to a map {{nowrap|1=<span style="text-decoration: overline;">''H''</span> : ''X'' &times; [0,&thinsp;1] &rarr; <span style="text-decoration: overline;">''Y''</span>}} such that {{nowrap|1=''p'' ○ <span style="text-decoration: overline;">''H''</span> = ''H''.}} The homotopy lifting property is used to characterize [[fibration]]s.

Another useful property involving homotopy is the [[homotopy extension property]],
which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with [[cofibration]]s.