Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological spaces.


But the usual definition of singular homology is on the category of topological spaces, and you can show that it is homotopy-invariant only after having defined it on the category of topological spaces.  

For example, the definition uses the free abelian group on the underlying set of the space of singular $n$-simplexes, and taking the underlying set of a space do not make sense in the homotopy category.


I would like to have a construction of singular homology that can be entirely carried out in the homotopy category.


I was thinking of categorifying the usual construction:  

Take the free spectrum ($(\infty,1)$ equivalent of abelian group?) on the space of singular n-simplexes, show that this is an "$(\infty,1)$ chain complex" and compute its "$(\infty,1)$ homology".


But I have the impression it will not work as is, because simplexes are only interesting as topological spaces, not as homotopy types.


**Is there a way to construct singular homology (or in fact any homology theory) directly in the homotopy category without using the category of topological spaces?**



Homology theory constructed in a homotopy-invariant way
Throughout the following, I'll say "homotopy category" when I really mean the weak homotopy category.


For a space $X$, the homology of $X$ is canonically isomorphic to the reduced homology of $X\_+$, which is $X$ with a disjoint basepoint. Therefore, it suffices to give a definition of the reduced homology of a based space.


The smash product $\wedge$ descends to a well-defined operation on the homotopy category of based topological spaces. For any $n \geq 0$, we have an object $K(\mathbb{Z},n)$ in the homotopy category, and weak equivalences $K(\mathbb{Z},n) \to \Omega K(\mathbb{Z},n+1)$ which are adjoint to maps $S^1 \wedge K(\mathbb{Z},n) \to K(\mathbb{Z},n+1)$. For any integer $m$, we can therefore define a directed system of sets
$$
[S^{m+k}, K(\mathbb{Z},k) \wedge X ] \to 
[S^{m+k + 1}, K(\mathbb{Z},k+1) \wedge X ] \to \cdots
$$
The colimit $colim\_k \pi\_{m+k}( K(\mathbb{Z},k) \wedge X)$ is isomorphic to the $m$'th reduced homology group of $X$ in a canonical way.


This kind of definition produces generalized homology theories, and this all falls into the subject of stable homotopy theory.


The method you suggest of taking the free spectrum in this $\infty$-categorical sense will give a homology theory, but instead of producing the homology groups of $X$ it will produce the stable homotopy groups. The "free abelian topological group" on the topological space $X$ can be used instead, and shown to give a good notion on the homotopy category; this will produce the homology of $X$, as a result of the work of Dold and Thom.


