2 edit based on Andrew Stacey's comments.

Corepresentable functors preserve products; homology does not.

One replacement is the following. Let X be a CW-complex with basepoint. Then the spaces {K(Z,n)} represent reduced integral homology in the sense that for sufficiently large n, the reduced homology Hk(X) coincides with the homotopy groups of the smash product:

pin+k(X ^ K(Z,n)) = [Sn+k, X ^ K(Z,n)]

This is some kind of "stabilization", and it factors through taking the n-fold suspension of X. Taking suspensions makes wedges more and more closely related to products, and makes . This doesn't make homology closer and closer to representable, but provides some alternative description that's more workable than simply an abstract functor.

1

Corepresentable functors preserve products; homology does not.

One replacement is the following. Let X be a CW-complex with basepoint. Then the spaces {K(Z,n)} represent reduced integral homology in the sense that for sufficiently large n, the reduced homology Hk(X) coincides with the homotopy groups of the smash product:

pin+k(X ^ K(Z,n)) = [Sn+k, X ^ K(Z,n)]

This is some kind of "stabilization", and it factors through taking the n-fold suspension of X. Taking suspensions makes wedges more and more closely related to products, and makes homology closer and closer to representable.