The derived category D(A) of an abelian category A (Verdier, 1960s) is constructed by taking chain complexes ... → A^{n-1} → A^n → A^{n+1} → ... and formally inverting quasi-isomorphisms (maps inducing isomorphisms on all cohomology groups H^n). This makes objects with the same cohomology isomorphic in D(A), even if they're distinct as complexes. Derived categories unify homological algebra: the derived functors (Ext, Tor, sheaf cohomology) become ordinary functors between derived categories. Kontsevich's Homological Mirror Symmetry conjecture (1994) says the derived category of coherent sheaves on a Calabi-Yau manifold is equivalent to the Fukaya category of its mirror — a profound geometric duality.