its chain homotopy equivalence class. Since every chain homotopy equivalence is a quasi-isomorphism, factors through this functor. Consequently can be equally well viewed as a localization of the homotopy category. From the point of view of model categories, the derived category D(A) is the true 'homotopy category' of the category of complexes, whereas K(A) might be called the 'naive homotopy category'. Constructing the derived category There are several possible constructions of the derived category. When is a small category, then there is a direct construction of the derived category by formally adjoining inverses of quasi-isomorphisms. This is an instance of the general construction of a category by generators and relations. When is a large category, this construction does not work for set theoretic reasons. This construction builds morphisms as equivalence classes of paths. If has a proper class of objects, all of which are isomorphic, then there is a proper class of paths between any two of these objects. The generators and relations construction therefore only guarantees that the morphisms between two objects form a proper class. However, the morphisms between two objects in a category are usually required to be sets, and so this construction fails to produce an actual category. Even when is small, however, the construction by generators and relations generally results in a category whose structure is opaque, where morphisms are arbitrarily long paths subject to a mysterious equivalence relation. For this reason, it is conventional to construct the derived category more concretely even when set theory is not at issue. These other constructions go through the homotopy category. The collection of quasi-isomorphisms in forms a multiplicative system. This is a collection of conditions that allow complicated paths to be rewritten as simpler ones. The Gabriel–Zisman theorem implies that localization at a multiplicative system has a simple description in terms of roofs. A morphism in may be described as a pair , where for some complex , is a quasi-isomorphism and is a chain homotopy equivalence class of morphisms. Conceptually, this represents . Two roofs are equivalent if they have a common overroof. Replacing chains of morphisms with roofs also enables the resolution of the set-theoretic issues involved in derived categories of large categories. Fix a complex and consider the category whose objects are quasi-isomorphisms in with codomain and whose morphisms are commutative diagrams. Equivalently, this is the category of objects over whose structure maps are