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.
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Comment #7527 by Samuel Tiersma on
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