Injective objects in Mor(Ab)

Question

Consider the abelian (Grothendieck) category $\mathcal{C} := \mathrm{Fun}(\{0<1\},\mathrm{Ab}) = \mathrm{Mor}(\mathrm{Ab})$. Objects are morphisms $(A \to B)$ of abelian groups, morphisms are commutative diagrams. Equivalently, this is the category of abelian sheaves on the Sierpinski space.

Question. How do injective objects in $\mathcal{C}$ look like?

Since injective sheaves are stable under restriction (use extension by zero), clearly $(A \to B)$ injective implies that $A$ is injective. But is this sufficient (probably not)? When $A,B$ are injective, is the same true for $(A \to B)$?

Answer 1

I will use notation $A_0 \to A_1$ for objects of $\mathrm{Mor}(\mathrm{Ab})$.

EDIT: previously I claimed something stronger (that I can produce lifting properties in the functor category without factorizations), but I am not so sure about it.

The following is a lot more general than necessary, but I think this added generality is also useful. Let $(\mathcal{L}, \mathcal{R})$ be a weak factorization system in a category $\mathcal{C}$ with enough colimits and limits for the following to make sense. Let $J$ be a Reedy category. Then in the functor category $\mathcal{C}^J$ the "Reedy $\mathcal{L}$-cofibrations" and "Reedy $\mathcal{R}$-fibrations" form a weak factorization system. By "Reedy $\mathcal{L}$-cofibrations" I mean morphisms of diagrams $X \to Y$ such that for every $j \in J$ the latching morphism $X_j \sqcup_{L_j X} L_j Y \to Y_j$ is in $\mathcal{L}$ and dually "Reedy $\mathcal{R}$-fibrations" are morphisms $X \to Y$ such that for every $j \in J$ the matching morphism $X_j \to M_j X \times_{M_j Y} Y_j$ is in $\mathcal{R}$. The proof is exactly as in the construction of the Reedy model structures and can be found for example in Hovey's Model Categories.

Now we take $\mathcal{C} = \mathrm{Ab}$, $\mathcal{L} = $ monomorphisms and $J = [1]$. Then $\mathcal{R}$ are split epimorphisms with injective kernel. The lifting properties are easily verified while the factorizations use the fact that there are enough injectives in $\mathrm{Ab}$. If $f : A \to B$ is a map in $\mathrm{Ab}$, pick an injective hull $i : A \to \hat A$, then $f$ factors as an injection $[i, f] : A \to \hat A \oplus B$ followed by a split surjection with injective kernel $\hat A \oplus B \to B$. We consider $J$ as a Reedy category where $0$ has degree $1$ and $1$ has degree $0$. Then "Reedy $\mathcal{L}$-cofibrations" are monomorphisms again, so an object $X$ is injective if and only if the map $X \to 0$ is a "Reedy $\mathcal{R}$-fibration" i.e. when both $X_1 \to 0$ and $X_0 \to X_1$ are split epimorphisms with injective kernel i.e. when $X_0 \to X_1$ is a split epimorphism with injective source.