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*}

3.10 Sets with group action

Let $G$ be a group. We denote $G\textit{-Sets}$ the “big” category of $G$-sets. For any ordinal $\alpha $, we denote $G\textit{-Sets}_\alpha $ the full subcategory of $G\textit{-Sets}$ whose objects are in $V_\alpha $. As a notion for size of a $G$-set we take $\text{size}(S) = \max \{ \aleph _0, |G|, |S|\} $ (where $|G|$ and $|S|$ are the cardinality of the underlying sets). As above we use the function $Bound(\kappa ) = \kappa ^{\aleph _0}$.

Lemma 3.10.1. With notations $G$, $G\textit{-Sets}_\alpha $, $\text{size}$, and $Bound$ as above. Let $S_0$ be a set of $G$-sets. There exists a limit ordinal $\alpha $ with the following properties:

  1. We have $S_0 \cup \{ {}_ GG\} \subset \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$.

  2. For any $S \in \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$ and any $G$-set $T$ with $\text{size}(T) \leq Bound(\text{size}(S))$, there exists a $S' \in \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$ that is isomorphic to $T$.

  3. For any countable diagram category $\mathcal{I}$ and any functor $F : \mathcal{I} \to G\textit{-Sets}_\alpha $, the limit $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} F$ and colimit $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} F$ exist in $G\textit{-Sets}_\alpha $ and are the same as in $G\textit{-Sets}$.

Proof. Omitted. Similar to but easier than the proof of Lemma 3.9.2 above. $\square$

Lemma 3.10.2. Let $\alpha $ be an ordinal as in Lemma 3.10.1 above. The category $G\textit{-Sets}_\alpha $ satisfies the following properties:

  1. The $G$-set ${}_ GG$ is an object of $G\textit{-Sets}_\alpha $.

  2. (Co)Products, fibre products, and pushouts exist in $G\textit{-Sets}_\alpha $ and are the same as their counterparts in $G\textit{-Sets}$.

  3. Given an object $U$ of $G\textit{-Sets}_\alpha $, any $G$-stable subset $O \subset U$ is isomorphic to an object of $G\textit{-Sets}_\alpha $.

Proof. Omitted. $\square$


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