What is the necessary and sufficient condition for abelian categories to have arbitrary direct limit?

Question

As a beginner to learn homological algebra, I have just learned about the direct system and its direct limit.As R-mod categories have arbitrary coproducts indexed by a set, any direct system in R-mod indexed by J has direct limit which is defined as the quotient mod of the direct product. Then，there comes a further question: Since R-mod category is a special abelian category with arbitrary direct limit and not all of abelian categories have this property, what is the necessary and sufficient condition ? And a different question is: when does a direct system in an abelian category has direct limit? A direct system in category C indexed by a quasi-ordered set J is a covariant functor from J to C.
It seems that an abelian category has arbitrary direct limit indexed by a set iff it has arbitrary coproducts. But I can't prove it. If this is right, could you give me some hints?

Answer 1

You are correct that every abelian category has coequalizers given by cokernels. So if the category has all small coproducts, it has all colimits (aka directed limits).

Another approach is the following: an abelian category has filtered colimits (aka filtered directed limits) iff it has all colimits.

Clearly, if it has all colimits, then it also has filtered colimits. Now let us prove the converse.

An abelian category has finite coproducts, given by direct sums. Then given a set of objects $\{A_i\}_{i\in I}$ of objects in our category, we have the coproduct as the filtered colimit of all finite coproducts of the $A_i$. Therefore, we have coproducts and coequalizers, hence all colimits.

The more interesting question is when a filtered colimit of exact sequences remains exact. This condition is part of the definition of Grothendieck categories (which also includes the existence of a generator).