Starting right from the defintions, what would be the shortest way to prove, that the category of abelian groups, $\mathcal{Ab}$, has exact directed limits (This means for every directed set $I$ is the colimit functor $$\operatorname{Colim}\colon\operatorname{Fun}(I,\mathcal{Ab})\rightarrow\mathcal{Ab}$$ an exact functor)?

Especially, I don't want to use, that exactness of directed limits is equivalent of some conditions on the lattice of subobjects of the category or that taking directed colimits in the category of sets commutes with finite limits.

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    $\begingroup$ The easiest way is to use the concrete descriptions of directed colimits in $\mathbf{Ab}$. If you're hoping to get an abstract nonsense proof, then let me point out that $\mathbf{Ab}^\mathrm{op}$ is also an abelian category, but directed colimits in $\mathbf{Ab}^\mathrm{op}$ are not exact. $\endgroup$ – Zhen Lin Jul 10 '13 at 21:43
  • $\begingroup$ Yeah you're right. Uisng the explicit description of directed colimits in $\mathcal{Ab}$ one can easily see, that taking directed colimits preserves monomorphisms and as the colimit functor is a left-adjoint, this suffices. Thanks. $\endgroup$ – John Jul 10 '13 at 21:52
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    $\begingroup$ Why don't you want to use the fact that taking directed colimits in $\text{Set}$ commutes with finite limits? It's clearly the reason this is true. $\endgroup$ – Qiaochu Yuan Jul 11 '13 at 5:27

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