2

I recently read the proof here that direct limits are exact functors in the category of $R$-modules. In that proof, I don't see where Arturo is using the fact that we are working in a category of modules, but in this question we get an example showing that taking direct limits is not necessarily exact in an arbitrary category.

So where are we using that we work in a category of modules in the proof that direct limits are exact functors?

stillconfused
  • 1,805

1 Answers1

0

Many of the arguments in Arturo's proof use elements. Some can likely be removed, e.g. the ones in the proof that the composition is $0$, but I don't see how to remove the $k$ from the line

so by exactness of the original diagram we know that there exists $k \in K_i$...

hunter
  • 29,847
  • I see. So would it be fair to say that the proof should work in any abelian category whose objects are sets? – stillconfused Sep 29 '22 at 22:46
  • 1
    That is a vague question. If I were feeling generous I would point out that if the forgetful functor preserves direct limits (= filtered colimits) and finite limits and is conservative then the answer is yes. If I were feeling malicious I would say that any concretisable abelian category not satisfying AB5 is a counterexample. – Zhen Lin Sep 29 '22 at 22:51