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In hopes of understanding algebra better, I've been reading here and there about protomodular categories and the like. Among other things, the theory surrounding these (and some other) categories gives a fuller hierarchy from pointed to abelian categories.

Looking back at the axioms of abelian categories, I'm suddenly wondering about the axiom $$\operatorname{Coker}\ker f\overset{\cong}{\longrightarrow}\operatorname{Ker}\operatorname{coker}f.$$ Drawing some pictures of (graphs of) pointed functions between pointed sets, it seems that the isomorphy of this arrow is capturing a sort of "subtractivity" property of the function: all injectivity information is stored in the kernel. That is, there cannot be a large fiber which is not "replicated" inside the kernel.

To explain what I mean, I'll try and give a useful example.

Example. Let $\bf n$ denote the pointed set $ \left\{ 0,\dots ,n-1 \right\}$ with $0$ as the basepoint. Looking at the pointed function $f:\mathbf 4\overset{\mod 2}{\longrightarrow} \mathbf 2$, one may see that $$\operatorname{Ker}f\cong \mathbf 2,\operatorname{Coker}\ker f\cong \mathbf 3,\operatorname{Coker}f\cong \mathbf 1,\operatorname{Ker}\operatorname{coker}f\cong \mathbf 2.$$ There is no isomorphism because the fiber of $1\in \mathbf 2$ is not visible in the fiber of $0\in \mathbf 2$.

Looking at the same function, now as a group homomorphism $g:\mathbb Z/4\mathbb Z\to \mathbb Z/2\mathbb Z$, we have $$\operatorname{Ker}g\cong \mathbf 2,\operatorname{Coker}\ker g\cong \mathbf 2,\operatorname{Coker}g\cong \mathbf 1,\operatorname{Ker}\operatorname{coker}g\cong \mathbf 2,$$ and the problem does not appear.

Thus my naive hope is that the condition $$\operatorname{Coker}\ker f\overset{\cong}{\longrightarrow}\operatorname{Ker}\operatorname{coker}f$$ is actually equivalent to $(\text{monic}\iff \text{zero kernel})$.

  1. Does $\operatorname{Coker}\ker f\overset{\cong}{\longrightarrow}\operatorname{Ker}\operatorname{coker}f$ even hold in (finitely complete) pointed protomodular categories?
  2. Is the above equivalence true for finitely complete pointed protomodular categories?
  3. Does this by any chance characterize such categories? (Doesn't make sense to expect this - already in groups the situation is much more structured since the subtraction operation makes all fibers isomorphic. This is way stronger (I think) than being injective away from the kernel.)

Remarks. By proposition 3.1.21 of this book1, finitely complete pointed protomodularity implies $(\text{monic}\iff \text{zero kernel})$. Also, I'm pretty sure that in the presence of enough (co)limits, $\operatorname{Coker}\ker f\overset{\cong}{\longrightarrow}\operatorname{Ker}\operatorname{coker}f$ is equivalent to $(\mathrm{StrEpi},\mathrm{StrMono})$ being an orthogonal factorization system.

1Borceux, Francis; Bourn, Dominique, Mal’cev, protomodular, homological and semi-abelian categories, Mathematics and its Applications (Dordrecht) 566. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1961-0/hbk). xiii, 479 p. (2004). ZBL1061.18001, MR2044291.

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