= Derived category of sheaves
{title2=$D(X)$}
The derived category of sheaves is formed by localizing cochain complexes of sheaves at quasi-isomorphisms, equivalently localizing their homotopy category. The bounded, bounded below and bounded above variants are denoted $D^b(X)$, $D^+(X)$ and $D^-(X)$. Cohomology sheaves are invariant under quasi-isomorphism and therefore define functors on these categories. Constructible bounded complexes form a natural subcategory on which <Verdier duality> reverses arrows and, under the usual finiteness hypotheses, is an involution.
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