One first defines a weak homotopy equivalence: a continuous map is called a weak homotopy equivalence if it induces a bijection on sets of path components and a bijection on homotopy groups with arbitrary base points. Then the (true) homotopy category is defined by localizing the category of topological spaces with respect to the weak homotopy equivalences. That is, the objects are still the topological spaces, but an inverse morphism is added for each weak homotopy equivalence. This has the effect that a continuous map becomes an isomorphism in the homotopy category if and only if it is a weak homotopy equivalence. There are obvious functors from the category of topological spaces to the naive homotopy category (as defined above), and from there to the homotopy category.