The Stacks project

Definition 6.4.4. Let $X$ be a topological space.

  1. A presheaf of abelian groups on $X$ or an abelian presheaf over $X$ is a presheaf of sets $\mathcal{F}$ such that for each open $U \subset X$ the set $\mathcal{F}(U)$ is endowed with the structure of an abelian group, and such that all restriction maps $\rho ^ U_ V$ are homomorphisms of abelian groups, see Lemma 6.4.3 above.

  2. A morphism of abelian presheaves over $X$ $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets which induces a homomorphism of abelian groups $\mathcal{F}(U) \to \mathcal{G}(U)$ for every open $U \subset X$.

  3. The category of presheaves of abelian groups on $X$ is denoted $\textit{PAb}(X)$.

Comments (0)

There are also:

  • 4 comment(s) on Section 6.4: Abelian presheaves

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Your email address will not be published.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 006K. Beware of the difference between the letter 'O' and the digit '0'.



View Definition 6.4.4 as pdf