In this post we shall see a natural example that should give some motivation for derived functors of functors on the homotopy category of cochain complexes in an abelian category $ \mathcal{A}$. At first glance, the formalism of derived functors in this setting may seem less intuitive than the formalism of classical derived functors, partially because of the need to consider hypercohomology (cohomology of complexes), but hopefully in this and in future posts we will see some examples that show the elegance of the theory.
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