Let $M$ be a smooth manifold, $A_M$ the de Rham algebra of $M$, $D_{A_M}$ the derived category of the category of differential graded (dg) $A_M$-modules and $D^+_c(M)$ the bounded below constructible derived category of sheaves of real vector spaces on $M$. The category $D^+_c(M)$ knows a lot about the topology of $M$; for example, it allows one to compute the real cohomology of $M$ together with all Massey products. However, unsurprisingly, this category is also quite complicated. Informally, the question I'd like to ask is: can one describe at least some pieces of $D^+_c(M)$ in terms of dg-modules (which is something much more manageable)?

In "Equivariant sheaves and functors", 12.3, J Bernstein and V. Lunts construct two mutually inverse equivalences $\gamma_M:\langle \mathbb{R}_M\rangle\to \langle A_M\rangle$ and $\mathcal{L}_M: \langle A_M\rangle\to \langle \mathbb{R}_M\rangle$. Here $\langle\cdot\rangle$ stands for the full triangulated subcategory generated by $\cdot$, $\mathbb{R}_M$ is the constant sheaf on $M$ and $\langle \mathbb{R}_M\rangle$ and $\langle A_M\rangle$ are subcategories of $D^+_c(M)$ and $D_{A_M}$ respectively.

The functors are defined as follows: $\gamma_M$ takes a sheaf, multiplies it by the de Rham complex and then takes the global sections; $\mathcal{L}_M$ takes a module, replaces it with a $\mathcal{K}$-projective resolution and multiplies the result by the de Rham complex. (A complex of dg $A_M$-modules is $\mathcal{K}$-projective, if $Hom$ from it to an acyclic complex is acyclic.)

Notice that $\gamma_M$ is in fact defined on the whole of $D^+(M)$. I would like to ask: is there a subcategory $D$ of $D^+_c(M)$ larger than the one generated by the constant sheaf such that $\gamma_M$ restricted to $D$ is fully faithful? In particular, if $i:N\subset M$ is a submanifold and $M,N,M\setminus N$ are all simply connected, what happens if we take $D=\langle\mathbb{R}_M, i_*\mathbb{R}_N\rangle$?

Here is a related result. Suppose we fix a stratification of $M$ with all strata and their closures simply-connected. Consider the subcategory $D\subset D^+_c(M)$ formed by complexes which are constructible with respect to the chosen stratification and let $I^*$ be an injective resolution of the direct sum of the constant sheaves on the strata. Then, due to a result by B. Keller (Deriving dg-algebras, Ann ENS, 1994, no 1, 63-102) by taking $C^*\to Hom(C^*,I^*)$ we get a fully faithful functor from $D$ to $D_{End(I^*)}$ where $End(I^*)$ is the (global) endomorphism algebra of $I^*$. However, this is not exactly what I'm looking for since the endomorphism algebra is still quite difficult to describe explicitly in the example I'm interested in.