Let $X$ and $Y$ be two topological spaces and $R$ be commutative ring. Let $D_c^b(X, R)$ and $D_c^b(Y,R)$ be their respective bounded derived categories of constructible sheaves of $R$-modules. I have three questions:
If $f:X\rightarrow Y$ is a weak homotopy equivalence does it induce (under any of the standard cohomological operations) an equivalence between $D_c^b(X, R)$ and $D_c^b(Y,R)$?
If 1. is not true in general, is there some interesting subcategory of topological spaces where it does become true? Perhaps CW-complexes?
If 1. is not true does it become so if we restrict to the full subcategories of $D_c^b(X, R)$ and $D_c^b(Y,R)$ whose objects have local system cohomology?
I suppose what I'm really asking is whether it is possible to naturally associate a bounded derived categories of constructible sheaves of $R$-modules to a homotopy type?