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

$\mathcal{A}$ has direct sums,

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

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

Here are the dual notions

$\mathcal{A}$ has products,

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

$\mathcal{A}$ 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.

## Comments (2)

Comment #7527 by Samuel Tiersma on

Comment #7658 by Stacks Project on

There are also: