# Inverse limit of epics in abelian category

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.

## 1 Answer

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.