# 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 $C_x = R\pi_* (\mathbb{Z}_X)$ 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?

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The map $X \mapsto C_X$ does not induce an injection on isomorphism classes. In order to apply th Whitehead theorem you need a map first. –  Torsten Ekedahl May 26 '10 at 16:29
First consider two such spaces whose cohomology is the same but whose ring structure is different, for example $\mathbb{C}P^n$ and a wedge of even spheres. The singular complexes are certainly isomorphic, but this could not be induced by a map of spaces. So your functor does not induce an injection on isomorphism classes.
To see that the functor is not faithful, consider the Hopf map, $S^3 \to S^2$, which surely induces the zero map in homology.