Editing Homotopy (section)

===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'').