ct.category theory - Topological homotopy category as derived category

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 $C_x = Rpi_* (mathbb{Z}_X)$
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?