The Stacks project

7.2 Presheaves

Let $\mathcal{C}$ be a category. A presheaf of sets is a contravariant functor $\mathcal{F}$ from $\mathcal{C}$ to $\textit{Sets}$ (see Categories, Remark 4.2.11). So for every object $U$ of $\mathcal{C}$ we have a set $\mathcal{F}(U)$. The elements of this set are called the sections of $\mathcal{F}$ over $U$. For every morphism $f : V \to U$ the map $\mathcal{F}(f) : \mathcal{F}(U) \to \mathcal{F}(V)$ is called the restriction map and is often denoted $f^\ast : \mathcal{F}(U) \to \mathcal{F}(V)$. Another way of expressing this is to say that $f^*(s)$ is the pullback of $s$ via $f$. Functoriality means that $g^* f^* (s) = (f \circ g)^*(s)$. Sometimes we use the notation $s|_ V := f^\ast (s)$. This notation is consistent with the notion of restriction of functions from topology because if $W \to V \to U$ are morphisms in $\mathcal{C}$ and $s$ is a section of $\mathcal{F}$ over $U$ then $s|_ W = (s|_ V)|_ W$ by the functorial nature of $\mathcal{F}$. Of course we have to be careful since it may very well happen that there is more than one morphism $V \to U$ and it is certainly not going to be the case that the corresponding pullback maps are equal.

Definition 7.2.1. A presheaf of sets on $\mathcal{C}$ is a contravariant functor from $\mathcal{C}$ to $\textit{Sets}$. Morphisms of presheaves are transformations of functors. The category of presheaves of sets is denoted $\textit{PSh}(\mathcal{C})$.

Note that for any object $U$ of $\mathcal{C}$ the functor of points $h_ U$, see Categories, Example 4.3.4 is a presheaf. These are called the representable presheaves. These presheaves have the pleasing property that for any presheaf $\mathcal{F}$ we have
\begin{equation} \label{sites-equation-map-representable-into-presheaf} \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) = \mathcal{F}(U). \end{equation}

This is the Yoneda lemma (Categories, Lemma 4.3.5).

Similarly, we can define the notion of a presheaf of abelian groups, rings, etc. More generally we may define a presheaf with values in a category.

Definition 7.2.2. Let $\mathcal{C}$, $\mathcal{A}$ be categories. A presheaf $\mathcal{F}$ on $\mathcal{C}$ with values in $\mathcal{A}$ is a contravariant functor from $\mathcal{C}$ to $\mathcal{A}$, i.e., $\mathcal{F} : \mathcal{C}^{opp} \to \mathcal{A}$. A morphism of presheaves $\mathcal{F} \to \mathcal{G}$ on $\mathcal{C}$ with values in $\mathcal{A}$ is a transformation of functors from $\mathcal{F}$ to $\mathcal{G}$.

These form the objects and morphisms of the category of presheaves on $\mathcal{C}$ with values in $\mathcal{A}$.

Remark 7.2.3. As already pointed out we may consider the category of presheaves with values in any of the “big” categories listed in Categories, Remark 4.2.2. These will be “big” categories as well and they will be listed in the above mentioned remark as we go along.

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