quasi-isomorphisms. Then the multiplicative system condition implies that the morphisms in from to are assuming that this colimit is in fact a set. While is potentially a large category, in some cases it is controlled by a small category. This is the case, for example, if is a Grothendieck abelian category (meaning that it satisfies AB5 and has a set of generators), with the essential point being that only objects of bounded cardinality are relevant. In these cases, the limit may be calculated over a small subcategory, and this ensures that the result is a set. Then may be defined to have these sets as its sets. There is a different approach based on replacing morphisms in the derived category by morphisms in the homotopy category. A morphism in the derived category with codomain being a bounded below complex of injective objects is the same as a morphism to this complex in the homotopy category; this follows from termwise injectivity. By replacing termwise injectivity by a stronger condition, one gets a similar property that applies even to unbounded complexes. A complex is K-injective if, for every acyclic complex , we have . A straightforward consequence of this is that, for every complex , morphisms in are the same as such morphisms in . A theorem of Serpé, generalizing work of Grothendieck and of Spaltenstein, asserts that in a Grothendieck abelian category, every complex is quasi-isomorphic to a K-injective complex with injective terms, and moreover, this is functorial. In particular, we may define morphisms in the derived category by passing to K-injective resolutions and computing morphisms in the homotopy category. The functoriality of Serpé's construction ensures that composition of morphisms is well-defined. Like the construction using roofs, this construction also ensures suitable set theoretic properties for the derived category, this time because these properties are already satisfied by the homotopy category. Derived Hom-Sets As noted before, in the derived category the hom sets are expressed through roofs, or valleys , where is a quasi-isomorphism. To get a better picture of what elements look like, consider an exact sequence We can use this to construct a morphism by truncating the complex above, shifting it, and using the obvious morphisms above. In particular, we have the picture where the bottom complex has concentrated in degree , the only non-trivial upward arrow is the equality morphism, and the only-nontrivial downward arrow is . This