Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with algebraic strata, though this is not my case. Consider the set of isomorphism classes of sheaves on $X$ constructible with respect to this stratification (say, with complex coefficients).
Question. Is there any natural topology on this set? Is there a modular interpretation of this set (in the algebraic case)?
For example if there is just a single stratum, the set of such sheaves is the set of isomorphism classes of representations of $\pi_1(X)$. The topology can be defined choosing generators and relations in $\pi_1(X)$.
Another version of this question is to consider instead of sheaves the set of isomorphism classes of objects of the bounded derived category of sheaves with cohomology sheaves constructible with respect to the given stratification.
Sorry if my question is too vague. I am not an expert in the field, and I am trying to figure out what is known, in order to understand what approximately I can hope for.