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:
We have $S_0 \cup \{ {}_ GG\} \subset \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$.
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 an $S' \in \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha )$ that is isomorphic to $T$.
For any countable index 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:
The $G$-set ${}_ GG$ is an object of $G\textit{-Sets}_\alpha $.
(Co)Products, fibre products, and pushouts exist in $G\textit{-Sets}_\alpha $ and are the same as their counterparts in $G\textit{-Sets}$.
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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