
## 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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