There is a functor from Top to hTop that sends spaces to themselves and morphisms to their homotopy classes.
[2] Example: The circle S1, the plane R2 minus the origin, and the Möbius strip are all homotopy equivalent, although these topological spaces are not homeomorphic.
That is, the objects are still the topological spaces, but an inverse morphism is added for each weak homotopy equivalence.
For example, for a weak homotopy equivalence of topological spaces f : X → Y, the associated homomorphism f* : Hi (X,Z) → Hi (Y,Z) of singular homology groups is an isomorphism for all natural numbers i.
In particular, two homotopic maps from X to Y induce the same homomorphism on singular homology groups.
Singular cohomology has an even better property: it is a representable functor on the homotopy category.
That is, for each abelian group A and natural number i, there is a CW complex K(A,i ) called an Eilenberg–MacLane space and a cohomology class u in H i(K(A,i ),A) such that the resulting function (giving by pulling u back to X) is bijective for all topological spaces X.
[6] Here [X,Y ] must be understood to mean the set of maps in the true homotopy category, if one wants this statement to hold for all topological spaces X.
The associated homotopy category is defined by localizing C with respect to the weak equivalences.
An important example is the standard model structure on simplicial sets: the associated homotopy category is equivalent to the homotopy category of topological spaces, even though simplicial sets are combinatorially defined objects that lack any topology.
Some topologists prefer instead to work with compactly generated weak Hausdorff spaces; again, with the standard model structure, the associated homotopy category is equivalent to the homotopy category of all topological spaces.
Then there is a model structure on the category of chain complexes of objects in A, with the weak equivalences being the quasi-isomorphisms.