This is going to be a long story, but let me try to explain how we are inevitably led to injective resolutions and derived functors if we want to study chain complexes up to "homology invariance" (= quasi-isomorphism).
Let $\mathcal{A}$ be an abelian category.
Let $\textbf{Ch} (\mathcal{A})$ be the category of chain complexes in $\mathcal{A}$.
I will use homological grading and write $\partial$ for the differential operator.
Since $\mathcal{A}$ is abelian, basic homological algebra can be applied to chain complexes in $\mathcal{A}$.
Suppose we are interested in studying chain complexes in $\mathcal{A}$ but we want to consider chain complexes the same if they have the same homology.
This leads us to define:
Definition.
A quasi-isomorphism of chain complexes in $\mathcal{A}$ is a morphism $f : A \to B$ in $\textbf{Ch} (\mathcal{A})$ such that, for every integer $n$, the induced morphism $\mathrm{H}_n (f) : \mathrm{H}_n (A) \to \mathrm{H}_n (B)$ of homology objects is an isomorphism in $\mathcal{A}$.
The derived category $\mathbf{D} (\mathcal{A})$ is the category obtained by freely inverting the quasi-isomorphisms in $\textbf{Ch} (\mathcal{A})$.
That means we have a functor $\gamma : \textbf{Ch} (\mathcal{A}) \to \mathbf{D} (\mathcal{A})$ that sends quasi-isomorphisms in $\textbf{Ch} (\mathcal{A})$ to isomorphisms in $\mathbf{D} (\mathcal{A})$, and every functor $\textbf{Ch} (\mathcal{A}) \to \mathcal{C}$ that sends quasi-isomorphisms in $\textbf{Ch} (\mathcal{A})$ to isomorphisms in $\mathcal{C}$ factors through the functor $\gamma : \textbf{Ch} (\mathcal{A}) \to \mathbf{D} (\mathcal{A})$ in a unique way.
For example, the homology functors $\textrm{H}_n : \textbf{Ch} (\mathcal{A}) \to \mathcal{A}$ certainly send quasi-isomorphisms to isomorphisms, so they induce functors $\mathbf{D} (\mathcal{A}) \to \mathcal{A}$.
But what are the morphisms of $\mathbf{D} (\mathcal{A})$?
It turns out that every morphism $\gamma P \to \gamma Q$ in $\mathbf{D} (\mathcal{A})$ is of the form $(\gamma j)^{-1} \circ \gamma f$ for some morphism $f : P \to \hat{Q}$ and quasi-isomorphism $j : Q \to \hat{Q}$ in $\textbf{Ch} (\mathcal{A})$.
Furthermore, $\mathbf{D} (\mathcal{A})$ has an additive structure and $\gamma : \textbf{Ch} (\mathcal{A}) \to \mathbf{D} (\mathcal{A})$ is additive.
So, for each $Q$ in $\textbf{Ch} (\mathcal{A})$, $P \mapsto \textrm{Hom}_{\mathbf{D} (\mathcal{A})} (\gamma P, \gamma Q)$ defines a functor $\textbf{Ch} (\mathcal{A})^\textrm{op} \to \textbf{Ab}$ that sends quasi-isomorphisms to isomorphisms, and $Q \mapsto (P \mapsto \textrm{Hom}_{\mathbf{D} (\mathcal{A})} (\gamma P, \gamma Q))$ defines a functor $\textbf{Ch} (\mathcal{A}) \to [\textbf{Ch} (\mathcal{A})^\textrm{op}, \textbf{Ab}]$ that sends quasi-isomorphisms to isomorphisms.
What is its relation to $Q \mapsto (P \mapsto \textrm{Hom}_{\textbf{Ch} (\mathcal{A})} (P, Q))$?
Definition.
Let $P$ and $Q$ be chain complexes in $\mathcal{A}$.
The chain complex $\textrm{Hom}_\mathcal{A} (P, Q)$ of abelian groups is defined as follows:
$$\begin{align}
\textrm{Hom}_\mathcal{A} (P, Q)_n & = \prod_m \textrm{Hom}_\mathcal{A} (P_m, Q_{m + n}) \\
(\partial (f))_m & = \partial \circ f_{m+1} - (-1)^n f_m \circ \partial
\end{align}$$
Observe that the 0-cycles of $\textrm{Hom}_\mathcal{A} (P, Q)$ are precisely the morphisms $P \to Q$ in $\textbf{Ch} (\mathcal{A})$, whereas the homology group $\mathrm{H}_0 (\textrm{Hom}_\mathcal{A} (P, Q))$ can be identified with the chain homotopy classes of morphisms $P \to Q$.
Anyway, varying $P$, we obtain a functor $\textrm{Hom}_\mathcal{A} (-, Q) : \textbf{Ch} (\mathcal{A})^\textrm{op} \to \textbf{Ch} (\textbf{Ab})$.
Unfortunately, it does not preserve quasi-isomorphisms in general, so $\mathrm{H}_0 (\textrm{Hom}_\mathcal{A} (P, Q))$ is different from $\textrm{Hom}_{\mathbf{D} (\mathcal{A})} (P, Q)$.
Definition.
A K-injective chain complex in $\mathcal{A}$ is a chain complex $Q$ in $\mathcal{A}$ such that $\textrm{Hom}_\mathcal{A} (-, Q) : \textbf{Ch} (\mathcal{A})^\textrm{op} \to \textbf{Ch} (\textbf{Ab})$ preserves quasi-isomorphisms.
Proposition.
Let $Q$ be a chain complex in $\mathcal{A}$.
The following are equivalent:
- $Q$ is a K-injective chain complex in $\mathcal{A}$.
- $\gamma \textrm{Hom}_\mathcal{A} (-, Q) : \textbf{Ch} (\mathcal{A})^\textrm{op} \to \mathbf{D} (\textbf{Ab})$ sends quasi-isomorphisms in $\textbf{Ch} (\mathcal{A})$ to isomorphisms in $\mathbf{D} (\textbf{Ab})$.
- $\mathrm{H}_0 (\textrm{Hom}_\mathcal{A} (-, Q)) : \textbf{Ch} (\mathcal{A})^\textrm{op} \to \textbf{Ab}$ sends quasi-isomorphisms in $\textbf{Ch} (\mathcal{A})$ to isomorphisms in $\textbf{Ab}$.
Proof.
Mostly straightforward.
The trick to get from statement 3 to statement 1 is to observe that
$$\mathrm{H}_n (\textrm{Hom}_\mathcal{A} (P, Q)) \cong \mathrm{H}_0 (\textrm{Hom}_\mathcal{A} (P [-n], Q))$$
where $P [-n]$ is the chain complex defined by $P [-n]_m = P_{m - n}$ with differential $P [-n]_m \to P [-n]_{m - 1}$ given by $(-1)^n$ times the differential $P_{m - n} \to P_{m - n - 1}$, and this isomorphism is natural in $P$ (and $Q$ too). ◼
In other words, a K-injective chain complex is a chain complex that "perceives" quasi-isomorphisms as chain homotopy equivalences.
Here is an instance of this phenomenon:
Lemma.
Let $g : Q \to P$ be a morphism in $\textbf{Ch} (\mathcal{A})$.
If $Q$ is K-injective and $g : Q \to P$ is a quasi-isomorphism, then there is a morphism $f : P \to Q$ in $\textbf{Ch} (\mathcal{A})$ such that $f \circ g : Q \to Q$ is chain homotopic to $\textrm{id}_Q$.
Proof.
The induced homomorphism $\mathrm{H}_0 (\textrm{Hom}_\mathcal{A} (P, Q)) \to \mathrm{H}_0 (\textrm{Hom}_\mathcal{A} (Q, Q))$ is an isomorphism.
In particular, the homology class of $\textrm{id}_Q$ has a preimage, say represented by a 0-cycle $f$ in $\textrm{Hom}_\mathcal{A} (P, Q)$, i.e. a morphism $P \to Q$ in $\textbf{Ch} (\mathcal{A})$.
But that means $g \circ f$ is in the same homology class as $\textrm{id}_Q$, and the homology classes of 0-cycles in $\textrm{Hom}_\mathcal{A} (Q, Q)$ are the chain homotopy classes of morphisms $Q \to Q$ in $\textbf{Ch} (\mathcal{A})$, so we are done. ◼
Corollary.
If both $P$ and $Q$ are K-injective chain complexes in $\mathcal{A}$, then the following are equivalent:
- $f : P \to Q$ is a quasi-isomorphism in $\textbf{Ch} (\mathcal{A})$.
- $f : P \to Q$ is a chain homotopy equivalence in $\textbf{Ch} (\mathcal{A})$. ◼
Why does this matter?
Well, for a general additive functor $F : \mathcal{A} \to \mathcal{B}$, the induced functor $\textbf{Ch} (F) : \textbf{Ch} (\mathcal{A}) \to \textbf{Ch} (\mathcal{B})$ is not guaranteed to preserve quasi-isomorphisms.
Indeed, $\textbf{Ch} (F)$ preserving quasi-isomorphisms is equivalent to $F$ preserving (short) exact sequences.
But $\textbf{Ch} (F)$ always preserves chain homotopy equivalences.
So if there were a "best approximation" of $\textbf{Ch} (F)$ by a functor that preserves quasi-isomorphisms, and every chain complex in $\mathcal{A}$ is quasi-isomorphic to a K-injective one, then the "best approximation" should be determined by the restriction of $\textbf{Ch} (F)$ to the full subcategory of K-injective chain complexes in $\mathcal{A}$.
Definition.
The right derived functor of a functor $G : \textbf{Ch} (\mathcal{A}) \to \mathcal{C}$ is a functor $\mathbf{R} G : \mathbf{D} (\mathcal{A}) \to \mathcal{C}$ equipped with a natural transformation $\eta : G \Rightarrow (\mathbf{R} G) \gamma$ such that, for every functor $H : \mathcal{C} \to \mathcal{D}$ and every functor $K : \mathbf{D} (\mathcal{A}) \to \mathcal{D}$ and every natural transformation $\phi : H G \Rightarrow K \gamma$, there is a unique natural transformation $\psi : H \mathbf{R} G \Rightarrow K$ such that $\psi \gamma \bullet \eta = \phi$.
(This is stronger than the original definition by Verdier.)
Theorem.
Let $G : \textbf{Ch} (\mathcal{A}) \to \mathcal{C}$ be a functor that sends chain homotopy equivalences in $\textbf{Ch} (\mathcal{A})$ to isomorphisms in $\mathcal{C}$.
Assume for every chain complex $P$ in $\mathcal{A}$ we have a quasi-isomorphism $j_P : P \to \hat{P}$ in $\textbf{Ch} (\mathcal{A})$ where $\hat{P}$ is a K-injective chain complex in $\mathcal{A}$.
Then the right derived functor $\mathbf{R} G : \textbf{D} (\mathcal{A}) \to \mathcal{C}$ exists and $\eta_Q : G Q \to (\mathbf{R} G) (\gamma Q)$ is an isomorphism for every K-injective chain complex $Q$ in $\mathcal{A}$.
Proof.
You can do this directly, but it is helpful to introduce the category $\mathbf{K} (\mathcal{A})$, which is $\textbf{Ch} (\mathcal{A})$ modulo chain homotopy.
In $\mathbf{K} (\mathcal{A})$, for every morphism $f : P \to Q$, there is a unique morphism $\hat{f} : \hat{P} \to \hat{Q}$ such that the following diagram commutes:
$$\require{AMScd}
\begin{CD}
P @>{j_P}>> \hat{P} \\
@V{f}VV @VV{\hat{f}}V \\
Q @>>{j_Q}> \hat{Q}
\end{CD}$$
(This is not always true in $\textbf{Ch} (\mathcal{A})$!)
Thus we may make $P \mapsto \hat{P}$ and $f \mapsto \hat{f}$ into a functor from $\mathbf{K} (\mathcal{A})$ to the full subcategory $\mathbf{K} (\mathcal{A})_\textrm{K-inj}$ of K-injective chain complexes (modulo chain homotopy).
Since $j_P : P \to \hat{P}$ is an isomorphism in $\mathbf{K} (\mathcal{A})$ if $P$ is K-injective, this exhibits $\mathbf{K} (\mathcal{A})_\textrm{K-inj}$ as a reflective subcategory of $\mathbf{K} (\mathcal{A})$.
Notice that $\hat{f} : \hat{P} \to \hat{Q}$ is an isomorphism in $\textbf{K} (\mathcal{A})_\textrm{K-inj}$ if $f : P \to Q$ is a quasi-isomorphism, so we get a functor $R : \mathbf{D} (\mathcal{A}) \to \textbf{K} (\mathcal{A})_\textrm{K-inj}$ where $R (\gamma P) = \hat{P}$ and $R (\gamma f) = \hat{f}$.
You can also show that $\mathbf{K} (\mathcal{A})$ is also the category obtained by freely inverting the chain homotopy equivalences in $\textbf{Ch} (\mathcal{A})$, so $G : \textbf{Ch} (\mathcal{A}) \to \mathcal{C}$ factors through the quotient $\textbf{Ch} (\mathcal{A}) \to \mathbf{K} (\mathcal{A})$, say as $\bar{G} : \mathbf{K} (\mathcal{A}) \to \mathcal{C}$.
Then you can check that taking $(\mathbf{R} G) = \bar{G} R$ and $\eta_P : G P \to (\mathbf{R} G) (\gamma P)$ to be $G j_P : G P \to G \hat{P}$ works. ◼
Hopefully the above convinces you that K-injective chain complexes are important.
But what are they, in more elementary terms?
Proposition.
Let $Q$ be a chain complex in $\mathcal{A}$ concentrated in degree 0.
The following are equivalent:
- $Q$ is a K-injective chain complex in $\mathcal{A}$.
- $Q_0$ is an injective object in $\mathcal{A}$.
Proof.
Essentially, $\textrm{Hom}_\mathcal{A} (-, Q) : \textbf{Ch} (\mathcal{A})^\textrm{op} \to \textbf{Ch} (\textbf{Ab})$ can be identified with $\textrm{Hom}_\mathcal{A} (-, Q_0): \textbf{Ch} (\mathcal{A})^\textrm{op} \to \textbf{Ch} (\textbf{Ab})$.
(There is a subtlety about the signs in the differentials.)
The latter preserves quasi-isomorphisms if and only if $\textrm{Hom}_\mathcal{A} (-, Q_0) : \mathcal{A}^\textrm{op} \to \textbf{Ab}$ is exact.
But that is the definition of injective object, so we are done. ◼
So the simplest examples of K-injective chain complexes are injective objects.
More generally:
Proposition.
Let $Q$ be a chain complex of injective objects in $\mathcal{A}$.
If $Q$ is bounded above (i.e. there is an integer $N$ such that $Q_n = 0$ for all $n \ge N$), then $Q$ is K-injective.
(Proof omitted.)
Unfortunately it is not true that K-injective chain complexes are chain complexes of injectives.
For example, for any object $A$ in $\mathcal{A}$,
$$\begin{CD}
\cdots @>>> 0 @>>> A @>{\textrm{id}}>> A @>>> 0 @>>> \cdots
\end{CD}$$
is a K-injective chain complex in $\mathcal{A}$, because the property of being K-injective is chain homotopy equivalence invariant, and $0$ is certainly a K-injective chain complex.
Nonetheless, I hope that this convinces you that injectivity is a natural notion in the context of homological algebra and not merely something that happens to work well.
