Algebraically constructible complexes of analytic field sheaves

Let $X$ be a separated finite-type complex algebraic variety, i.e. a classical algebraic variety over $\mathbb{C}$ (Classical algebraic prevarieties, regular maps, and varieties), regarded as a topological space with its analytic topology (Analytic topology and finite smooth algebraic stratifications of complex varieties), and let $k$ be a field. Write ${\mathrm{Sh}}_k(X)$ for the category of sheaves of $k$-vector spaces and $k$-linear morphisms (A sheaf on a topological space, Presheaves and sheaves of groups, rings, and modules, Modules on a ringed space); it is an abelian category (Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories), so its bounded derived category $D^b({\mathrm{Sh}}_k(X))$ is defined with the conventions of Derived category of an abelian category, under the standing smallness hypothesis recorded there.

The category $D_c^b(X;k)$ is the full subcategory of $D^b({\mathrm{Sh}}_k(X))$ consisting of the complexes $A$ satisfying:

1. $A$ is cohomologically bounded: $H^j(A)=0$ for $\mathrm{\mid }j\mathrm{\mid }$ sufficiently large; and
2. $A$ is algebraically constructible: there exists a finite smooth algebraic stratification with the frontier condition $\mathfrak{S}$ of $X$ (Analytic topology and finite smooth algebraic stratifications of complex varieties) such that for every stratum $S\in \mathfrak{S}$ and every integer $j$ the restriction $H^j(A){\mathrm{\mid }}_S$ is locally constant of finite rank (Locally constant sheaves of finite rank).

These are topological constructible sheaves, not coherent or quasi-coherent sheaves. We use the shift convention of Derived category of an abelian category: $H^j(A[m])=H^{j+m}(A)$. A groupoid local system is identified with a locally constant sheaf only through an explicit local comparison, which is not made here.

Constructibility is a condition on the cohomology sheaves of a complex, and it is quantified existentially over finite smooth algebraic frontier stratifications: no stratification is fixed, and no existence or common-refinement theorem for stratifications is asserted by this definition. Shifting replaces $H^j$ by $H^{j+m}$, which preserves cohomological boundedness and shifts a witnessing stratification nowhere, so the class is closed under shifts and under isomorphisms in $D^b({\mathrm{Sh}}_k(X))$. No choice principle is used.