Editing Homotopy (section)

== Properties ==

=== 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.

===Groups===

{{main|Homotopy group}}
Since the relation of two functions <math>f, g\colon X\to Y</math> being homotopic relative to a subspace is an equivalence relation, we can look at the [[equivalence class]]es of maps between a fixed ''X'' and ''Y''. If we fix <math>X = [0,1]^n</math>, the unit interval [0,&thinsp;1] [[cartesian product|crossed]] with itself ''n'' times, and we take its [[Boundary (topology)|boundary]] <math>\partial([0,1]^n)</math> as a subspace, then the equivalence classes form a group, denoted <math>\pi_n(Y,y_0)</math>, where <math>y_0</math> is in the image of the subspace <math>\partial([0,1]^n)</math>.

We can define the action of one equivalence class on another, and so we get a group. These groups are called the [[homotopy group]]s. In the case <math>n = 1</math>, it is also called the [[fundamental group]].

===Homotopy category===
{{Main|Homotopy category}}
The idea of homotopy can be turned into a formal category of [[category theory]]. The '''[[homotopy category]]''' is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces ''X'' and ''Y'' are isomorphic in this category if and only if they are homotopy-equivalent.  Then a [[functor]] on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.

For example, homology groups are a ''functorial'' homotopy invariant: this means that if ''f'' and ''g'' from ''X'' to ''Y'' are homotopic, then the [[group homomorphism]]s induced by ''f'' and ''g'' on the level of [[homology group]]s are the same: H<sub>''n''</sub>(''f'') = H<sub>''n''</sub>(''g'') : H<sub>''n''</sub>(''X'') → H<sub>''n''</sub>(''Y'') for all ''n''.  Likewise, if ''X'' and ''Y'' are in addition [[connectedness|path connected]], and the homotopy between ''f'' and ''g'' is pointed, then the group homomorphisms induced by ''f'' and ''g'' on the level of [[homotopy group]]s are also the same: π<sub>''n''</sub>(''f'') = π<sub>''n''</sub>(''g'') : π<sub>''n''</sub>(''X'') → π<sub>''n''</sub>(''Y'').