TITLE: Topological homotopy category as derived category 
QUESTION [7 upvotes]: 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?

REPLY [7 votes]: More generally one can ask: What kind of topological homotopy categories are related to "algebraic" derived-categories?

First, to have a triangulated category analogous to $\mathcal{D}(Mod-\mathbb{Z})$, you should replace $Ho( SimplicialSpaces )$ by $Ho( Spectra )$, i.e. the stable version of the former. This said, there are intrinsic obstructions to express $Ho( Spectra )$ as the derived category of an algebraic one (even a "stable" category). Fo full details, see
Muro, F.; Schwede, S.; Strickland, N.: 
Triangulated categories without models. Invent. Math. 170 (2007), no. 2, 231–241.
and
Schwede, S.:
Algebraic versus topological triangulated categories. Triangulated categories, 389–407, 
London Math. Soc. Lecture Note Ser., 375, Cambridge Univ. Press, Cambridge, 2010.