Regarding condition $AB5$

Question

My question is regarding the condition $AB5$ for an abelian category $\mathcal{A}$ i.e. direct sums exists and filtered colimits are exact. Now taking colimit is right exact in an abelian category given the colimit exists. Say I'm given monomorphisms $\phi_{i}: M_{i} \rightarrow N_{i}$ indexed over filtered category $I$. The kernel is a finite limit. Now, I know filtered colimit commutes with finite limit when the target category is $Sets$. Then does it mean when the target category is an arbitrary abelian category, then filtered colimit need not commute with kernel? More generally, I would like to have some idea on what kind of target category does this commutativity hold?

Answer 1

I am not sure whether I understand your question, but for abelian categories, AB5 is equivalent to the fact that filtered colimits preserve finite limits.

First of all, a filtered colimit functor, or any colimit functor, on an abelian category is always right exact (since it is a left adjoint to the diagonal embedding/constant diagram functor). It follows that when a filtered colimit functor is not exact, i.e. AB5 fails, the failure of exactness is due to the failure of left exactness, i.e. failure to preserve kernels.

(In a sense, this occurs in nature quite often, i.e. take $\mathcal{A}=(\mathrm{Mod-}R)^{op}$. Then this is typically not an AB5-category since inverse limits are typically not exact in $\mathrm{Mod-}R$.)

On the other hand, in an AB5-abelian category, filtered colimit functors preserve all finite limits. The reason is that one can represent all limits as a composition of simpler ones. Typically, all limits are obtained just as equalizer of pair of morphisms between certain products (whose form depends only on the diagram shape), and in the case of finite limits in an Abelian category, any finite limit is obtained as a kernel of a map between some finite biproducts, i.e. coproducts. But again, filtered colimit functor always preserves coproducts (it is a left adjoint), so preserving finite limits again comes down to preserving kernels, which is guaranteed by exactness.