In the Introduction of his Derived Categories for the working mathematician Richard Thomas mentions the following theorem of Whitehead.

Suppose that $X,Y$ are simplicial complexes, then the underlying topological spaces |X|, |Y| (both simply connected) are homotopy equivalent if and only if there are maps of simplicial complexes $X \leftarrow Z \rightarrow X$ inducing quasi-isomorphims $C_X \leftarrow C_Z \rightarrow C_Y$ of simlicial-chain complexes.

This suggest that there is a relation between the homotopy category of simplicial complexes and the derived cagtegory of abelian groups. E.g. the functor

$ Ho( SimplicialSpaces ) \longrightarrow \mathcal{D}(Mod-\mathbb{Z}), X \mapsto C_X$.

induces an injection on the level of isomorphy classs when restricted to simply-connected spaces.

Does this functor have any good properties? What about homomorphisms (fullness/faithfulness)? How does it help to study topological spaces?

More generally one can ask: What kind of topological homotopy categories are related to "algebraic" derived-categories?

The above example can be interpreted as $C_x = R\pi_* (\mathbb{Z}_X)$ where $\pi: X \rightarrow \{pt\}$ is the projetion to a point, and $\mathbb{Z}-Mod$ occures as abelian sheaves on $pt$. One could try to generalize using sheaved spaces $X \rightarrow B$ over a scheme $B$.

This seem to be quiet obvious questions. However I have seen nobody elaborating on this so far. Or maybe I am missing a link to something well known?

notinduce an injection on isomorphism classes. In order to apply th Whitehead theorem you need a map first. $\endgroup$