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Tag 001O

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

Example 4.3.4. Functor of points. For any $U\in \mathop{\rm Ob}\nolimits(\mathcal{C})$ there is a contravariant functor $$ \begin{matrix} h_U & : & \mathcal{C} & \longrightarrow & \textit{Sets} \\ & & X & \longmapsto & \mathop{\rm Mor}\nolimits_\mathcal{C}(X, U) \end{matrix} $$ which takes an object $X$ to the set $\mathop{\rm Mor}\nolimits_\mathcal{C}(X, U)$. In other words $h_U$ is a presheaf. Given a morphism $f : X\to Y$ the corresponding map $h_U(f) : \mathop{\rm Mor}\nolimits_\mathcal{C}(Y, U)\to \mathop{\rm Mor}\nolimits_\mathcal{C}(X, U)$ 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 presheaf associated to $U$. If $\mathcal{C}$ is the category of schemes this functor is sometimes referred to as the functor of points of $U$.

    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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