I'm trying to get some intuition for AB5 categories and Grothendieck categories by asking primitive questions.

First of all, why ask for exact filtered colimits? Are they there simply to have some pleasant properties of algebraic categories like $R$-$\mathsf{Mod}$, or are they there to interact with some specific aspects of the abelian structure?

Second of all, why ask for a generator? This paper provides several characterizations of generators in $R$-$\mathsf{Mod}$, but they don't really mean anything to me.

This MSE question, for instance, really makes me feel AB5 categories have a lot more going on than just abelian + resembling algebraic category. It doesn't seem there's any way for a general AB5 categoriy to magically inherit this distributivity of intersections directly from $\mathsf{Set}$. On the other hand, the wonderfully detailed proof given by Makoto Kato has me pretty lost in all the structure.

References to any English document containing (intuitive) discussion of these axioms is most welcome.


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