TITLE: Meaning of injectives objects in a category
QUESTION [6 upvotes]: I'm struggling to understand the meaning/motivation behind injective objects in (abelian) categories, especially in the context of group cohomology. They seem to be mostly mysterious as one mostly cares about having enough of them. Having enough injectives is great since we can define derived functors and derived functors are  useful. But this reason for defining them seems kind of backwards.
I am wondering if there is any "intrinsic" interest / reason in defining and studying injectives objects, which would naturally lead to looking at injective resolutions and defining derived functors.

REPLY [2 votes]: 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.