The Stacks project

Definition 19.10.1. Let $\mathcal{A}$ be an abelian category. We name some conditions

1. $\mathcal{A}$ has direct sums,

2. $\mathcal{A}$ has AB3 and direct sums are exact,

3. $\mathcal{A}$ has AB3 and filtered colimits are exact.

Here are the dual notions

1. $\mathcal{A}$ has products,

2. $\mathcal{A}$ has AB3* and products are exact,

3. $\mathcal{A}$ has AB3* and cofiltered limits are exact.

We say an object $U$ of $\mathcal{A}$ is a generator if for every $N \subset M$, $N \not= M$ in $\mathcal{A}$ there exists a morphism $U \to M$ which does not factor through $N$. We say $\mathcal{A}$ is a Grothendieck abelian category if it has AB5 and a generator.