Motivation for Definition of Derived Category

The quasi-isomorphisms in the category of chain complexes are related to homotopy theory, and in a sense localizing at the quasi-isomorphisms is like localizing topological spaces at the homotopy equivalences. It is primarily a computational tool for chain complexes and resolutions of various objects.

The purpose achieved by localizing, in general, is that you obtain a new category in which what was previously only weak equivalences (or quasi-isomorphisms or whatever you want to call them) are now actual isomorphisms. The story is as follows.

In a category you always have a notion of isomorphisms. Quite often though you have a category but you also have a notion of equivalency which is weaker than that described by isomorphisms. In other words, you have a bunch of morphisms which included the isomorphisms, and you think of objects for which such an arrow exists between them as essentially the same. For instance, in $Top$ isomorphism corresponds to homeomorphism, but being homotopy equivalent is strictly weaker. For a topologist, working in $Top$ is quite fine. For a homotopy theorist, working in $Top$ is not so fine, since it isn't really the right category for the homotopy theorist to work in - its notion of isomorphic objects is too strict. So, the homotopy theorist really wants a category of topological spaces where homotopy equivalences are actually the isomorphisms in the category. In other categories similar situations arise.

The most obvious way to turn all the weak equivalences into actual isomorphisms is to turn them into isomorphisms, by brute force. This is technically difficult though and results in poorly manageable categories. There are various techniques for localizing a category that can assist in producing something more manageable.