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*}

Example 7.6.5. Let $G$ be a group. Consider the category $G\textit{-Sets}$ whose objects are sets $X$ with a left $G$-action, with $G$-equivariant maps as the morphisms. An important example is ${}_ GG$ which is the $G$-set whose underlying set is $G$ and action given by left multiplication. This category has fiber products, see Categories, Section 4.7. We declare $\{ \varphi _ i : U_ i \to U\} _{i\in I}$ to be a covering if $\bigcup _{i\in I} \varphi _ i(U_ i) = U$. This gives a class of coverings on $G\textit{-Sets}$ which is easily seen to satisfy conditions (1), (2), and (3) of Definition 7.6.2. The result is not a site since both the collection of objects of the underlying category and the collection of coverings form a proper class. We first replace by $G\textit{-Sets}$ by a full subcategory $G\textit{-Sets}_\alpha $ as in Sets, Lemma 3.10.1. After this the site $(G\textit{-Sets}_\alpha , \text{Cov}_{\kappa , \alpha '}(G\textit{-Sets}_\alpha ))$ gotten by suitably restricting the collection of coverings as in Sets, Lemma 3.11.1 will be denoted $\mathcal{T}_ G$.

As a special case, if the group $G$ is countable, then we can let $\mathcal{T}_ G$ be the category of countable $G$-sets and coverings those jointly surjective families of morphisms $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ such that $I$ is countable.


Comments (2)

Comment #2992 by Dario WeiƟmann on

Typo: "...which is easily see ... " should read "...which is easily seen..."

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  • 2 comment(s) on Section 7.6: Sites

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