Topological homotopy category as derived category

Question

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?

Answer 1

I don't think this is actually an injection on isomorphism classes. We can see this by restricting to extremely nice spaces, say finite simply-connected CW complexes with free homology. In this situation cohomology and homology are dual and we have a Whitehead theorem for cohomology--it makes things easy to think about, though I'm sure you can concoct counterexamples with weaker hypotheses.

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.

Similarly, one sees that the functor is not full, there are maps that fail to respect the ring structure and so could not be induced by any map of spaces, one example is the map that identifies the homologies of the two spaces above.

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.

Answer 2

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.