Verdier showed how derived functors associated with an abelian category A can be viewed as Kan extensions along embeddings of A into suitable derived categories [Mac Lane]. Derived equivalence It may happen that two abelian categories A and B are not equivalent, but their derived categories D(A) and D(B) are. Often this is an interesting relation between A and B. Such equivalences are related to the theory of t-structures in triangulated categories. Here are some examples. Let be an abelian category of coherent sheaves on the projective line over a field k. Let K2-Rep be an abelian category of representations of the Kronecker quiver with two vertices. They are very different abelian categories, but their (bounded) derived categories are equivalent. Let Q be any quiver and P be a quiver obtained from Q by reversing some arrows. In general, the categories of representations of Q and P are different, but Db(Q-Rep) is always equivalent to Db(P-Rep). Let X be an abelian variety, Y its dual abelian variety. Then Db(Coh(X)) is equivalent to Db(Coh(Y)) by the theory of Fourier–Mukai transforms. Varieties with equivalent derived categories of coherent sheaves are sometimes called Fourier–Mukai partners. See also Homotopy category of chain complexes Derived noncommutative algebraic geometry Coherent sheaf cohomology Coherent duality Derived algebraic geometry Notes References Four textbooks that discuss derived categories are: Homological algebra Categories in category theory