## 59.9 Presheaves

A reference for this section is Sites, Section 7.2.

Definition 59.9.1. Let $\mathcal{C}$ be a category. A presheaf of sets (respectively, an abelian presheaf) on $\mathcal{C}$ is a functor $\mathcal{C}^{opp} \to \textit{Sets}$ (resp. $\textit{Ab}$).

Terminology. If $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, then elements of $\mathcal{F}(U)$ are called sections of $\mathcal{F}$ over $U$. For $\varphi : V \to U$ in $\mathcal{C}$, the map $\mathcal{F}(\varphi ) : \mathcal{F}(U) \to \mathcal{F}(V)$ is called the restriction map and is often denoted $s \mapsto s|_ V$ or sometimes $s \mapsto \varphi ^*s$. The notation $s|_ V$ is ambiguous since the restriction map depends on $\varphi$, but it is a standard abuse of notation. We also use the notation $\Gamma (U, \mathcal{F}) = \mathcal{F}(U)$.

Saying that $\mathcal{F}$ is a functor means that if $W \to V \to U$ are morphisms in $\mathcal{C}$ and $s \in \Gamma (U, \mathcal{F})$ then $(s|_ V)|_ W = s |_ W$, with the abuse of notation just seen. Moreover, the restriction mappings corresponding to the identity morphisms $\text{id}_ U : U \to U$ are the identity.

The category of presheaves of sets (respectively of abelian presheaves) on $\mathcal{C}$ is denoted $\textit{PSh} (\mathcal{C})$ (resp. $\textit{PAb} (\mathcal{C})$). It is the category of functors from $\mathcal{C}^{opp}$ to $\textit{Sets}$ (resp. $\textit{Ab}$), which is to say that the morphisms of presheaves are natural transformations of functors. We only consider the categories $\textit{PSh}(\mathcal{C})$ and $\textit{PAb}(\mathcal{C})$ when the category $\mathcal{C}$ is small. (Our convention is that a category is small unless otherwise mentioned, and if it isn't small it should be listed in Categories, Remark 4.2.2.)

Example 59.9.2. Given an object $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$, we consider the rule

$\begin{matrix} U & \longmapsto & h_ X(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, X) \\ V \xrightarrow {\varphi } U & \longmapsto & (\psi \mapsto \psi \circ \varphi ) : h_ X(U) \to h_ X(V). \end{matrix}$

This defines a functor $h_ X : \mathcal{C}^{opp} \to \textit{Sets}$ and hence a presheaf. This is called the representable presheaf associated to $X$. It is not true that representable presheaves are sheaves in every topology on every site.

Lemma 59.9.3 (Yoneda). Let $\mathcal{C}$ be a category, and $X, Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. There is a natural bijection

$\begin{matrix} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, Y) & \longrightarrow & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})} (h_ X, h_ Y) \\ \psi & \longmapsto & h_\psi = \psi \circ - : h_ X \to h_ Y. \end{matrix}$

Proof. See Categories, Lemma 4.3.5. $\square$

Comment #1389 by sdf on

In the terminology subsubsection, the $\mathcal{F}(\varphi)$ is going the wrong way since contravariant out of $\mathcal{F}$/covariant out of $\mathcal{F}^{\mathrm{opp}}$?

Comment #1391 by sdf on

Above comment should say "...contravariant out of $\mathcal{C}$/covariant out of $\mathcal{C}^{\mathrm{opp}}$?"

Comment #8391 by Francisco García on

There is something wrong with Example 59.9.2. Do we consider $V \xrightarrow {\varphi} U$ in $\mathcal{C}^{opp}$ or in $\mathcal{C}$? Assuming the morphism in $\mathcal{C}^{opp}$ I think both the composition $\varphi \circ -$ and the map $h_X(U) \to h_X(V)$ have to be switched. Compare with tag 4.3.4.

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