The Stacks project

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 presheaf associated to $U$. If $\mathcal{C}$ is the category of schemes this functor is sometimes referred to as the functor of points of $U$.


Comments (0)

There are also:

  • 6 comment(s) on Section 4.3: Opposite Categories and the Yoneda Lemma

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Your email address will not be published.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 001O. Beware of the difference between the letter 'O' and the digit '0'.