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
Comments (0)