The Stacks project

Example 7.6.5. Let $G$ be a group. Consider the category $G\textit{-Sets}$ whose objects are sets $X$ with a left $G$-action, with $G$-equivariant maps as the morphisms. An important example is ${}_ GG$ which is the $G$-set whose underlying set is $G$ and action given by left multiplication. This category has fiber products, see Categories, Section 4.7. We declare $\{ \varphi _ i : U_ i \to U\} _{i\in I}$ to be a covering if $\bigcup _{i\in I} \varphi _ i(U_ i) = U$. This gives a class of coverings on $G\textit{-Sets}$ which is easily seen to satisfy conditions (1), (2), and (3) of Definition 7.6.2. The result is not a site since both the collection of objects of the underlying category and the collection of coverings form a proper class. We first replace by $G\textit{-Sets}$ by a full subcategory $G\textit{-Sets}_\alpha $ as in Sets, Lemma 3.10.1. After this the site $(G\textit{-Sets}_\alpha , \text{Cov}_{\kappa , \alpha '}(G\textit{-Sets}_\alpha ))$ gotten by suitably restricting the collection of coverings as in Sets, Lemma 3.11.1 will be denoted $\mathcal{T}_ G$.

As a special case, if the group $G$ is countable, then we can let $\mathcal{T}_ G$ be the category of countable $G$-sets and coverings those jointly surjective families of morphisms $\{ \varphi _ i : U_ i \to U\} _{i \in I}$ such that $I$ is countable.