The Stacks project

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

\[ \begin{matrix} h_ X : & \mathcal{C}^{opp} & \longrightarrow & \textit{Sets} \\ & U & \longmapsto & h_ X(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, X) \\ & V \xrightarrow {\varphi } U & \longmapsto & \varphi \circ - : h_ X(U) \to h_ X(V). \end{matrix} \]

It is a presheaf, called the representable presheaf associated to $X$. It is not true that representable presheaves are sheaves in every topology on every site.

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View Example 59.9.2 as pdf