Let $F : \mathcal A \to \mathcal B$ be an additive functor between abelian categories. If $F$ is left exact, then under certain conditions $F$ admits right derived functors which “measure” its failure to be exact. If $F$ is right exact, then dually $F$ can often be derived.
Being left or right exact are ways to ber “almost exact, but not quite”. Another way to be “almost exact, but not quite” is to preserve both epimorphisms and monomorphisms — or equivalently, to preserve images. I’m tempted to call such a functor “outer exact”. For example, for fixed $n \in \mathbb Z$, the functor $Ab \to Ab, X \to nX$, which sends $X$ to the image of the map $n : X \to X$, is a functor which preserves images.
Question: Let $F : \mathcal A \to \mathcal B$ be an additive functor between nice abelian categories, and assume that $F$ preserves images. Then is there some collection of functors $I_n F : \mathcal A \to \mathcal B$ which “measure” the failure of $F$ to be exact in some sense?
This is of course a vague question, and probably most of the difficulty lies in trying to make the question more precise.
Motivation:
Let $\mathcal A$ be an abelian category, and let $\mathcal A^{[1]}$ be the category of morphisms in $\mathcal A$. Then I can think of three canonical functors $\mathcal A^{[1]} \to \mathcal A$: one can take the kernel, the cokernel, or the image. I know how to measure the failure of exactness of the first two, but not the third. There’s at least one context where these three functors sit in a kind of “triality” — if $\mathcal A$ is the Freyd envelope of a triangulated category, then images, kernels, and cokernels can be understood in terms of one another by rotating triangles. This leads me to hope that whatever I can do with kernels and cokernels, I can also do with images.
