diagram of complexes defines a morphism in the derived category. One application of this observation is the construction of the Atiyah-class. Remarks For certain purposes (see below) one uses bounded-below ( for ), bounded-above ( for ) or bounded ( for ) complexes instead of unbounded ones. The corresponding derived categories are usually denoted D+(A), D−(A) and Db(A), respectively. If one adopts the classical point of view on categories, that there is a set of morphisms from one object to another (not just a class), then one has to give an additional argument to prove this. If, for example, the abelian category A is small, i.e. has only a set of objects, then this issue will be no problem. Also, if A is a Grothendieck abelian category, then the derived category D(A) is equivalent to a full subcategory of the homotopy category K(A), and hence has only a set of morphisms from one object to another. Grothendieck abelian categories include the category of modules over a ring, the category of sheaves of abelian groups on a topological space, and many other examples. Composition of morphisms, i.e. roofs, in the derived category is accomplished by finding a third roof on top of the two roofs to be composed. It may be checked that this is possible and gives a well-defined, associative composition. Since K(A) is a triangulated category, its localization D(A) is also triangulated. For an integer n and a complex X, define the complex X[n] to be X shifted down by n, so that with differential By definition, a distinguished triangle in D(A) is a triangle that is isomorphic in D(A) to the triangle X → Y → Cone(f) → X[1] for some map of complexes f: X → Y. Here Cone(f) denotes the mapping cone of f. In particular, for a short exact sequence in A, the triangle X → Y → Z → X[1] is distinguished in D(A). Verdier explained that the definition of the shift X[1] is forced by requiring X[1] to be the cone of the morphism X → 0. By viewing an object of A as a complex concentrated in degree zero, the derived category D(A) contains A as a full subcategory. Morphisms in the derived category include information about all Ext groups: for any objects X and Y in A and any integer j, Projective and injective resolutions One can easily show that a