 Derived category of sheaves

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.