Are constructible derived categories invariant up to weak homotopy equivalence? 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?
 A: I don't know much about these things but I think 1. fails even if $X$ is a point and $Y$ is a line. Let's say also $R$ is a field. Then the category of constructible sheaves on $X$ is just $R$-Vect which is in particular semisimple, but the category of constructible sheaves on $Y$ is not semisimple: if we choose a sheaf on a point $p$ and on the complement $U$ of that point, then there are in general many non-isomorphic choices of a sheaf on $Y$ with given restrictions to $p$ and $U$ which fit in a short exact sequence. 
So their derived categories are not equivalent either.

Addendum: Maybe I can say something about how the usual homotopy invariance of cohomology is visible in the 'six functor' formalism. For any space $X$ let $\pi_X$ be the projection to a point, then the cohomology of $X$ (with any coefficients) just corresponds to functor $R \pi_{X\ast} \circ \pi_X^\ast$. If $f \colon X \to Y$ is an arbitrary map, then note that $\pi_Y \circ f = \pi_X$ which implies 
$$ R\pi_{Y\ast} \circ Rf_\ast \circ f^\ast \circ \pi_Y^\ast = R \pi_{X\ast} \circ \pi_X^\ast;$$
and now the morphism $\mathbf 1 \to Rf_\ast \circ f^\ast$ coming from the adjunction gives a map $f^\#  \colon R \pi_{Y\ast} \circ \pi_Y^\ast \to R \pi_{X\ast} \circ \pi_X^\ast$. Of course evaluating this equation on a choice of coefficients we get the usual map $H^\bullet(Y) \to H^\bullet(X)$. The correct way to express homotopy invariance is now that if $f$ and $g$ are homotopic maps $X \to Y$, then $f^\#$ and $g^\#$ are equal. This can be generalized to the relative situation, when $X$ and $Y$ are spaces over some base space $S$ and we consider the derived pushforward to $S$ instead of to a point, and we take homotopies over $S$.
