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The Stacks project

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 an S' \in \mathop{\mathrm{Ob}}\nolimits (G\textit{-Sets}_\alpha ) that is isomorphic to T.

  3. 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


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