4 reordered

One conceptual reason is, in technical terms, that "the derived category of (bounded-below) complexes is isomorphic to the category of (bounded-below) injective complexes." In less fancy language:

Say

First, when working with a single sheaf A, we can make it into a complex A with one A and everything else 0:

...→0→A→0→0→0→....

Then an "injective resolution" of A is really a complex of injectives I with a quasi-isomorphism A→I (a map of complexes which induces an isomorphism on cohomology).

Now, say A and B are any (bounded below) complexes of sheaves which have the same (cochain) cohomology. There may not be a map from one to the other giving rise to the isomorphism of their cohomologies (kernels mod image), i.e. a quasi-isomorphism (qis).

However, you can find complexes of injectives I and J, and maps a: A → I, b: B → J, and f: I → J such that a,b are qis, and f is a homotopy equivalence (in particular a qis). So, "as far as cohomology is concerned", you can replace A by I and B by J.

The "big picture" reason for this is that injectives are "flexible" in terms of extending maps into them (that's how they're defined), which is what allows the construction of the maps in the previous paragraph. Having , and having maps between things is good, because maps transform nicely under the application of functors.

When working with a single sheaf A rather than a complex of sheaves, we just make it into a complex A with one A and everything else 0:

...→0→A→0→0→0→....

Then an "injective resolution" of A is really a complex of injectives I with a quasi-isomorphism A→I.

3 arrows were backwards

One conceptual reason is, in technical terms, that "the derived category of (bounded-below) complexes is isomorphic to the category of (bounded-below) injective complexes." In less fancy language:

Say A and B are (bounded below) complexes of sheaves which have the same (cochain) cohomology. There may not be a map from one to the other giving rise to the isomorphism of their cohomologies (kernels mod image), i.e. a quasi-isomorphism (qis).

However, you can find complexes of injectives I and J, and maps a: IAAI, b: JBAJ, and f: I → J such that a,b are qis, and J f is a homotopy equivalence (in particular a qis). So, "as far as cohomology is concerned", you can replace A by I and B by J.

The "big picture" reason for this is that injectives are "flexible" in terms of extending maps into them (that's how they're defined), which is what allows the construction of the maps in the previous paragraph. Having maps between things is good, because maps transform nicely under the application of functors.

When working with a single sheaf A rather than a complex of sheaves, we just make it into a complex A with one A and everything else 0:

...→0→A→0→0→0→....

Then an "injective resolution" of A is really a complex of injectives that is quasi-isomorphic to I with a quasi-isomorphism A→I.

2 typo

One conceptual reason is, in technical terms, that "the derived category of (bounded-below) complexes is isomorphic to the category of (bounded-below) injective complexes." In less fancy language:

Say A and B are (bounded below) complexes of sheaves which have the same (cochain) cohomology. There may not be a map from one to the other giving rise to the isomorphism of their cohomologies (kernels mod image), i.e. a quasi-isomorphism (qis).

However, you can find complexes of injectives I and J, and maps a: I → A, b: J → A, and f: I → J such that a,b are qis, and J is a homotopy equivalence (in particular a qis). So, "as far as cohomology is concerned", you can replace A by AI and B by J.

When working with a single sheaf A rather than a complex of sheaves, we just make it into a complex A with one A and everything else 0:

...→0→A→0→0→0→....

Then an "injective resolution" of A is really a complex of injectives that is quasi-isomorphic to A.

1