The Stacks project

6.3 Presheaves

Definition 6.3.1. Let $X$ be a topological space.

  1. A presheaf $\mathcal{F}$ of sets on $X$ is a rule which assigns to each open $U \subset X$ a set $\mathcal{F}(U)$ and to each inclusion $V \subset U$ a map $\rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V)$ such that $\rho ^ U_ U = \text{id}_{\mathcal{F}(U)}$ and whenever $W \subset V \subset U$ we have $\rho ^ U_ W = \rho ^ V_ W \circ \rho ^ U_ V$.

  2. A morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of sets on $X$ is a rule which assigns to each open $U \subset X$ a map of sets $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ compatible with restriction maps, i.e., whenever $V \subset U \subset X$ are open the diagram

    \[ \xymatrix{ \mathcal{F}(U) \ar[r]^\varphi \ar[d]^{\rho ^ U_ V} & \mathcal{G}(U) \ar[d]^{\rho ^ U_ V} \\ \mathcal{F}(V) \ar[r]^\varphi & \mathcal{G}(V) } \]

    commutes.

  3. The category of presheaves of sets on $X$ will be denoted $\textit{PSh}(X)$.

The elements of the set $\mathcal{F}(U)$ are called the sections of $\mathcal{F}$ over $U$. For every $V \subset U$ the map $\rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V)$ is called the restriction map. We will use the notation $s|_ V := \rho ^ U_ V(s)$ if $s\in \mathcal{F}(U)$. This notation is consistent with the notion of restriction of functions from topology because if $W \subset V \subset U$ and $s$ is a section of $\mathcal{F}$ over $U$ then $s|_ W = (s|_ V)|_ W$ by the property of the restriction maps expressed in the definition above.

Another notation that is often used is to indicate sections over an open $U$ by the symbol $\Gamma (U, -)$ or by $H^0(U, -)$. In other words, the following equalities are tautological

\[ \Gamma (U, \mathcal{F}) = \mathcal{F}(U) = H^0(U, \mathcal{F}). \]

In this chapter we will not use this notation, but in others we will.

Definition 6.3.2. Let $X$ be a topological space. Let $A$ be a set. The constant presheaf with value $A$ is the presheaf that assigns the set $A$ to every open $U \subset X$, and such that all restriction mappings are $\text{id}_ A$.


Comments (0)


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Your email address will not be published.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