## Tag `001O`

Chapter 4: Categories > Section 4.3: Opposite Categories and the Yoneda Lemma

Example 4.3.4. Functor of points. For any $U\in \mathop{\mathrm{Ob}}\nolimits(\mathcal{C})$ there is a contravariant functor $$ \begin{matrix} h_U & : & \mathcal{C} & \longrightarrow & \textit{Sets} \\ & & X & \longmapsto & \mathop{Mor}\nolimits_\mathcal{C}(X, U) \end{matrix} $$ which takes an object $X$ to the set $\mathop{Mor}\nolimits_\mathcal{C}(X, U)$. In other words $h_U$ is a presheaf. Given a morphism $f : X\to Y$ the corresponding map $h_U(f) : \mathop{Mor}\nolimits_\mathcal{C}(Y, U)\to \mathop{Mor}\nolimits_\mathcal{C}(X, U)$ takes $\phi$ to $\phi\circ f$. We will always denote this presheaf $h_U : \mathcal{C}^{opp} \to \textit{Sets}$. It is called the

representable presheafassociated to $U$. If $\mathcal{C}$ is the category of schemes this functor is sometimes referred to as thefunctor of pointsof $U$.

The code snippet corresponding to this tag is a part of the file `categories.tex` and is located in lines 465–495 (see updates for more information).

```
\begin{example}
\label{example-hom-functor}
Functor of points.
For any $U\in \Ob(\mathcal{C})$ there is a contravariant
functor
$$
\begin{matrix}
h_U & : & \mathcal{C}
&
\longrightarrow
&
\textit{Sets} \\
& &
X
&
\longmapsto
&
\Mor_\mathcal{C}(X, U)
\end{matrix}
$$
which takes an object $X$ to the set
$\Mor_\mathcal{C}(X, U)$. In other words $h_U$ is a presheaf.
Given a morphism $f : X\to Y$ the corresponding map
$h_U(f) : \Mor_\mathcal{C}(Y, U)\to \Mor_\mathcal{C}(X, U)$
takes $\phi$ to $\phi\circ f$. We will always denote
this presheaf $h_U : \mathcal{C}^{opp} \to \textit{Sets}$.
It is called the {\it representable presheaf} associated to $U$.
If $\mathcal{C}$ is the category of schemes this functor is
sometimes referred to as the
\emph{functor of points} of $U$.
\end{example}
```

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