I'm going to make a radical suggestion (and hence I have marked this community wiki), that there is a rewarding way to study derived categories as the localizations of "categories with weak equivalences", or more specifically, homotopical categories in the sense of Dwyer-Hirschhorn-Kan-Smith. That is, any functor preserving weak equivalences between homotopical categories descends uniquely to localizations (by the universal property of localizations), so derived functors become the images of homotopical approximations of functors between the original categories (where homotopical approximation means in some sense the closest weak-equivalence preserving functor from the left or the right).

Indeed, using the simplicial localization of Dwyer-Kan (hammock localization or its more modern variant "Grothendieck-style simplicial localization" detailed in sections 34 and 35 of Dwyer-Hirschhorn-Kan-Smith (*Homotopy Limit Functors on Model and Homotopical categories*), we can embed the case of homotopical categories into the case of simplicially enriched categories. This turns our ordinary derived categories into categories enriched in weak homotopy types of CW complexes such that $\pi_0Map_C(X,Y):=Hom_{Ho(Set_\Delta)}(\Delta^0, Map_C(X,Y))\cong Hom_{W^{-1}C}(X,Y)$, where $W^{-1}C$ is the derived category. The papers of Dwyer-Kan (or the recollection of these techniques in Chapter 17 of Phil Hirschhorn's *Model Categories and their Localizations*) give a way to compute the mapping space in terms of simplicial and cosimplicial resolutions. This formulation is interesting because it lets us use the well-developed techniques of homotopical algebra in a simplicially-enriched category as well (for instance, there is a very mature theory of homotopy limits and colimits in this setting). This also gives the correct data and bypasses the need of working with triangulated categories, for instance. To use the language of the nLab (which is often scarier-sounding than it actually is), this is the $(\infty,1)$-categorical approach to derived categories.

(Note that in the above, we would let $C$ be an appropriate category of chain complexes (of sheaves)).

If I remember correctly, this approach is roughly discussed in SGA 4 exposé 2.

language. This is used widely both in algebraic geometry and in various areas of representation theory (some purely algebraic). So different sources will have different emphases when it comes to motivation or applications. $\endgroup$samemap as the edge map in the Cech to derived-functor spectral sequence? Try to prove it;I found this very difficult to prove for myself when I first learned these things (several pages of gigantic diagrams, etc.; maybe I was missing something obvious). Once I learned derived categories, it became a 2-line argument. $\endgroup$6more comments