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 (2)

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'.

There are also:

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

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 007V. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 6.16.3 as pdf