The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

7.40 Weakly contractible objects

A weakly contractible object of a site is one that satisfies the equivalent conditions of the following lemma.

Lemma 7.40.1. Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. The following conditions are equivalent

  1. For every covering $\{ U_ i \to U\} $ there exists a map of sheaves $h_ U^\# \to \coprod h_{U_ i}^\# $ right inverse to the sheafification of $\coprod h_{U_ i} \to h_ U$.

  2. For every surjection of sheaves of sets $\mathcal{F} \to \mathcal{G}$ the map $\mathcal{F}(U) \to \mathcal{G}(U)$ is surjective.

Proof. Assume (1) and let $\mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves of sets. For $s \in \mathcal{G}(U)$ there exists a covering $\{ U_ i \to U\} $ and $t_ i \in \mathcal{F}(U_ i)$ mapping to $s|_{U_ i}$, see Definition 7.11.1. Think of $t_ i$ as a map $t_ i : h_{U_ i}^\# \to \mathcal{F}$ via (7.12.3.1). Then precomposing $\coprod t_ i : \coprod h_{U_ i}^\# \to \mathcal{F}$ with the map $h_ U^\# \to \coprod h_{U_ i}^\# $ we get from (1) we obtain a section $t \in \mathcal{F}(U)$ mapping to $s$. Thus (2) holds.

Assume (2) holds. Let $\{ U_ i \to U\} $ be a covering. Then $\coprod h_{U_ i}^\# \to h_ U^\# $ is surjective (Lemma 7.12.4). Hence by (2) there exists a section $s$ of $\coprod h_{U_ i}^\# $ mapping to the section $\text{id}_ U$ of $h_ U^\# $. This section corresponds to a map $h_ U^\# \to \coprod h_{U_ i}^\# $ which is right inverse to the sheafification of $\coprod h_{U_ i} \to h_ U$ which proves (1). $\square$

Definition 7.40.2. Let $\mathcal{C}$ be a site.

  1. We say an object $U$ of $\mathcal{C}$ is weakly contractible if the equivalent conditions of Lemma 7.40.1 hold.

  2. We say a site has enough weakly contractible objects if every object $U$ of $\mathcal{C}$ has a covering $\{ U_ i \to U\} $ with $U_ i$ weakly contractible for all $i$.

  3. More generally, if $P$ is a property of objects of $\mathcal{C}$ we say that $\mathcal{C}$ has enough $P$ objects if every object $U$ of $\mathcal{C}$ has a covering $\{ U_ i \to U\} $ such that $U_ i$ has $P$ for all $i$.

The small ├ętale site of $\mathbf{A}^1_\mathbf {C}$ does not have any weakly contractible objects. On the other hand, the small pro-├ętale site of any scheme has enough contractible objects.


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