'set-sized' criterion for injectivity in an abelian category

Question

To detect an injective object in $R-\mathbf{Mod}$, it suffices to test for only set-sized collection of objects, by Baer's criterion. How do we do this for an arbitrary abelian category? Stacks project says that for any abelian category with enough injectives, we can find a small abelian full subcategory with enough injectives, including the desired set-sized collection of objects, such that the inclusion functor is exact, and conserves and reflects injectives. I came up with a way to achieve this without finding the general set-sized injectivity criterion. The desired category is usually constructed by making intermediate steps $X_0, X_1, \ldots$ and taking the union. For each step $X_n$, we can append the witnesses for non-injectivity of non-injective objects of $X_n$ to the next step $X_{n + 1}$. Apart from this, is there any general set-sized injectivity criterion? so that we could just include them at first without appending the witnesses.

Answer 1

In general there is no set of objects that suffices to test injectivity.

Let $\mathcal{C}$ be the category of functors from ordinals to abelian groups that are nonzero only on a set of ordinals. I.e., an object $F$ assigns an abelian group $F(\alpha)$ to each ordinal $\alpha$ and a homomorphism $f_{\alpha,\beta}:F(\alpha)\to F(\beta)$ for each pair of ordinals $\alpha\leq\beta$ such that $f_{\beta,\gamma}f_{\alpha,\beta}=f_{\alpha,\gamma}$ whenever $\alpha\leq\beta\leq\gamma$, and such that there is some $\alpha$ such that $\beta\geq\alpha\Rightarrow F(\beta)=0$. And a morphism $F\to G$ is a collection of homomorphisms $F(\alpha)\to G(\alpha)$ such that the obvious squares commute.

Then $\mathcal{C}$ is an abelian category (locally small because of the restriction on when $F(\alpha)\neq0$).

It is easy to see that the functor $S_{\alpha}$, where $S_{\alpha}(\alpha)=\mathbb{Z}$ and $S_{\alpha}(\beta)=0$ for $\beta\neq\alpha$, is not injective. But if you pick any set $\mathcal{F}$ of objects, then there is some ordinal $\alpha$ such that $F(\beta)=0$ for every $F\in\mathcal{F}$ and every $\beta\geq\alpha$. So there are no nonzero morphisms $F\to S_{\alpha}$ for $F\in\mathcal{F}$, and so the fact that $S_{\alpha}$ is not injective can't be detected using only the objects of $\mathcal{F}$.

After posting the example above, I remembered hearing about some rather interesting related results involving less contrived categories than the one above.

It follows from Lemma 2.5 in this recent paper of Šaroch and Trlifaj that if $R$ is a non-perfect ring, then it is independent of ZFC (the usual axioms of set theory) whether in the category of $R$-modules there is a set of epimorphisms that suffice to test projectivity. [Actually, this was proved in a much earlier paper of Trlifaj, but the statement in the paper I've linked to is less technical.]

This means that it is independent of ZFC whether the opposite category of the category of abelian groups answers the question in the OP!