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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 homomorphisms induced by f and g on the level of homology groups are the same: H (f) = H (g) : H (X) → H (Y) for all n. Likewise, if X and Y are in addition 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 groups are also the same: π (f) = π (g) : π (X) → π (Y).