The Stacks project

Lemma 6.16.3. Let $X$ be a topological space.

  1. Epimorphisms (resp. monomorphisms) in the category of presheaves are exactly the surjective (resp. injective) maps of presheaves.

  2. Epimorphisms (resp. monomorphisms) in the category of sheaves are exactly the surjective (resp. injective) maps of sheaves, and are exactly those maps which are surjective (resp. injective) on all the stalks.

  3. The sheafification of a surjective (resp. injective) morphism of presheaves of sets is surjective (resp. injective).

Proof. Omitted. $\square$

Comments (3)

Comment #6251 by Yiyang Wang on

Typo (though nothing serious) : lemma 6. 16. 3, (2),  '... and are exactly those maps with are Surjective...,  here with should be 'which'.

Comment #9510 by Branislav Sobot on

Isn't it a little bit weird to talk about sheafification in this lemma, since you introduce it only after this subsection?

There are also:

  • 2 comment(s) on Section 6.16: Exactness and points

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