The Stacks project

Example 7.33.7. Consider the site $\mathcal{T}_ G$ described in Example 7.6.5 and Section 7.9. The forgetful functor $u : \mathcal{T}_ G \to \textit{Sets}$ commutes with products and fibred products and turns coverings into surjective families. Hence it defines a point of $\mathcal{T}_ G$. We identify $\mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G)$ and $G\textit{-Sets}$. The stalk functor

\[ p^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) = G\textit{-Sets} \longrightarrow \textit{Sets} \]

is the forgetful functor. The pushforward $p_*$ is the functor

\[ \textit{Sets} \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{T}_ G) = G\textit{-Sets} \]

which maps a set $S$ to the $G$-set $\text{Map}(G, S)$ with action $g \cdot \psi = \psi \circ R_ g$ where $R_ g$ is right multiplication. In particular we have $p^{-1}p_*S = \text{Map}(G, S)$ as a set and the maps $S \to \text{Map}(G, S) \to S$ of Lemma 7.32.9 are the obvious ones.

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