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Does the AB4 axiom or AB5 axiom of abelian categories by grothendieck give the fact that cofiltered limits of epics are epic? Specifically, suppose $A_n$ maps surjectively to $A_{n-1}$ for each $n \in \mathbb{Z}$ does the limit map surjectively onto each $A_n$?

Motivation: trying to prove acyclic assembly lemma over abelian categories.

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Let $\mathsf{Ab}$ be the category of abelian groups, and $\mathsf{TAb}$ the full subcategory of torsion abelian groups.

Then $\mathsf{TAb}$ is an AB5 abelian subcategory of $\mathsf{Ab}$, and the inclusion reflects colimits (i.e., the colimit in $\mathsf{Ab}$ of any diagram of torsion abelian groups is torsion, and is also the colimit in $\mathsf{TAb}$.

$\mathsf{TAb}$ also has limits, but these are not the same as in $\mathsf{Ab}$, since a limit of torsion groups may not be a torsion group. To construct a limit in $\mathsf{TAb}$ you take the limit in $\mathsf{Ab}$ and take the torsion subgroup of that.

Now consider the diagram $$\dots\to\mathbb{Z}/p^3\mathbb{Z}\to\mathbb{Z}/p^2\mathbb{Z}\to\mathbb{Z}/p\mathbb{Z}$$ for some prime $p$, with all the maps surjective.

The limit in $\mathsf{Ab}$ is the group $\mathbb{Z}_p$ of $p$-adic integers, which is torsion free, and so its torsion subgroup (the limit in $\mathsf{TAb}$) is zero, so the natural maps to the groups $\mathbb{Z}/p^n\mathbb{Z}$ are certainly not epimorphisms.

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