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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.

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If you triangulate your space refining the stratification, a constructible sheaf is given by the data of a vector space $V_{\sigma}$ (a stalk at the barycenter, say) on each simplex $\sigma$ and a restriction map $V_{\sigma} \to V_{\tau}$ for any pair of incident simplices $\sigma \subset \tau$. If you fix all those vector spaces ("frame those stalks"), that defines a point in $\prod_{\sigma,\tau \mid \sigma \subset \tau} \mathrm{Hom}(V_{\sigma},V_{\tau})$.

The restriction maps are subject to some conditions: the triangle of restriction maps associated to a triple $\sigma \subset \tau \subset \upsilon$ should commute (cutting out a closed subset of $\prod$), and $V_{\sigma} \to V_{\tau}$ should be an isomorphism if the relative interiors of $\sigma$ and $\tau$ belong to the same stratum (cutting out an open subset of that closed subset).

That space depends on the triangulation, but the quotient (as a set or as a stack) by the action of $\prod \mathrm{GL}(V_{\sigma})$ doesn't.

The situation for sheaves of chain complexes is worse, even of chain complexes themselves (sheaves on a point). If you fix the chain groups and vary the differential, a sequence of acyclic complexes can converge to a non-acyclic complex of any size. If you want to consider the "moduli of quasi-isomorphism classes of sheaves" you have to decide what to make of this (use a stability condition, or make a derived moduli space, or there could be other options); maybe it depends on taste and circumstances.

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  • $\begingroup$ Many thanks. Is there a reference to this kind of material? $\endgroup$ – MKO Mar 20 '16 at 10:47
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    $\begingroup$ Hi Semyon. How about chapter VIII, section 1 of Kashiwara and Schapira's "Sheaves on Manifolds"? $\endgroup$ – David Treumann Mar 20 '16 at 15:11

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