One motivation for these categories is that many invariants of topological spaces are defined on the naive homotopy category or even on the true homotopy category. For example, for a weak homotopy equivalence of topological spaces "f": "X" → "Y", the associated homomorphism "f"*: "H""i"("X",Z) → "H""i"("Y",Z) of singular homology groups is an isomorphism for all natural numbers "i". It follows that, for each natural number "i", singular homology "H""i" can be viewed as a functor from the homotopy category to the category of abelian groups. In particular, two homotopic maps from "X" to "Y" induce the "same" homomorphism on singular homology groups.