From 3caae897aeb00e786ae128a5291d18152d30d9df Mon Sep 17 00:00:00 2001 From: Aise Johan de Jong Date: Wed, 17 Apr 2024 14:18:51 -0400 Subject: [PATCH] Fix confusion about representable presheaves MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Thanks to Francisco García https://stacks.math.columbia.edu/tag/03NB#comment-8391 --- CONTRIBUTORS | 1 + etale-cohomology.tex | 13 +++++++------ 2 files changed, 8 insertions(+), 6 deletions(-) diff --git a/CONTRIBUTORS b/CONTRIBUTORS index 7eafebe8..86ac0c40 100644 --- a/CONTRIBUTORS +++ b/CONTRIBUTORS @@ -141,6 +141,7 @@ Pierre-Yves Gaillard Juan Sebastian Gaitan Lennart Galinat Martin Gallauer +Francisco García Luis Garcia Xu Gao Toby Gee diff --git a/etale-cohomology.tex b/etale-cohomology.tex index cbccc382..abe2f004 100644 --- a/etale-cohomology.tex +++ b/etale-cohomology.tex @@ -457,16 +457,17 @@ \section{Presheaves} \begin{example} \label{example-representable-presheaf} -Given an object $X \in \Ob(\mathcal{C})$, we consider the functor +Given an object $X \in \Ob(\mathcal{C})$, we consider the rule $$ \begin{matrix} -h_X : & \mathcal{C}^{opp} & \longrightarrow & \textit{Sets} \\ -& U & \longmapsto & h_X(U) = \Mor_\mathcal{C}(U, X) \\ -& V \xrightarrow{\varphi} U & \longmapsto & -\varphi \circ - : h_X(U) \to h_X(V). +U & \longmapsto & h_X(U) = \Mor_\mathcal{C}(U, X) \\ +V \xrightarrow{\varphi} U & \longmapsto & +(\psi \mapsto \psi \circ \varphi) : h_X(U) \to h_X(V). \end{matrix} $$ -It is a presheaf, called the {\it representable presheaf associated to $X$.} +This defines a functor $h_X : \mathcal{C}^{opp} \to \textit{Sets}$ +and hence a presheaf. This is called the +{\it representable presheaf associated to $X$.} It is not true that representable presheaves are sheaves in every topology on every site. \end{example}