## 59.12 The example of G-sets

Let $G$ be a group and define a site $\mathcal{T}_ G$ as follows: the underlying category is the category of $G$-sets, i.e., its objects are sets endowed with a left $G$-action and the morphisms are equivariant maps; and the coverings of $\mathcal{T}_ G$ are the families $\{ \varphi _ i : U_ i \to U\} _{i\in I}$ satisfying $U = \bigcup _{i\in I} \varphi _ i(U_ i)$.

There is a special object in the site $\mathcal{T}_ G$, namely the $G$-set $G$ endowed with its natural action by left translations. We denote it ${}_ G G$. Observe that there is a natural group isomorphism

In particular, for any presheaf $\mathcal{F}$, the set $\mathcal{F}({}_ G G)$ inherits a $G$-action via $\rho $. (Note that by contravariance of $\mathcal{F}$, the set $\mathcal{F}({}_ G G)$ is again a left $G$-set.) In fact, the functor

is an equivalence of categories. Its quasi-inverse is the functor $X \mapsto h_ X$. Without giving the complete proof (which can be found in Sites, Section 7.9) let us try to explain why this is true.

If $S$ is a $G$-set, we can decompose it into orbits $S = \coprod _{i\in I} O_ i$. The sheaf axiom for the covering $\{ O_ i \to S\} _{i\in I}$ says that

\[ \xymatrix{ \mathcal{F}(S) \ar[r] & \prod _{i\in I} \mathcal{F}(O_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{i, j \in I} \mathcal{F}(O_ i \times _ S O_ j) } \]is an equalizer. Observing that fibered products in $G\textit{-Sets}$ are induced from fibered products in $\textit{Sets}$, and using the fact that $\mathcal{F}(\emptyset )$ is a $G$-singleton, we get that

\[ \prod _{i, j \in I} \mathcal{F}(O_ i \times _ S O_ j) = \prod _{i \in I} \mathcal{F}(O_ i) \]and the two maps above are in fact the same. Therefore the sheaf axiom merely says that $\mathcal{F}(S) = \prod _{i\in I} \mathcal{F}(O_ i)$.

If $S$ is the $G$-set $S= G/H$ and $\mathcal{F}$ is a sheaf on $\mathcal{T}_ G$, then we claim that

\[ \mathcal{F}(G/H) = \mathcal{F}({}_ G G)^ H \]and in particular $\mathcal{F}(\{ *\} ) = \mathcal{F}({}_ G G)^ G$. To see this, let's use the sheaf axiom for the covering $\{ {}_ G G \to G/H \} $ of $S$. We have

\begin{eqnarray*} {}_ G G \times _{G/H} {}_ G G & \cong & G \times H \\ (g_1, g_2) & \longmapsto & (g_1, g_1 g_2^{-1}) \end{eqnarray*}is a disjoint union of copies of ${}_ G G$ (as a $G$-set). Hence the sheaf axiom reads

\[ \xymatrix{ \mathcal{F} (G/H) \ar[r] & \mathcal{F}({}_ G G) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{h\in H} \mathcal{F}({}_ G G) } \]where the two maps on the right are $s \mapsto (s)_{h \in H}$ and $s \mapsto (hs)_{h \in H}$. Therefore $\mathcal{F}(G/H) = \mathcal{F}({}_ G G)^ H$ as claimed.

This doesn't quite prove the claimed equivalence of categories, but it shows at least that a sheaf $\mathcal{F}$ is entirely determined by its sections over ${}_ G G$. Details (and set theoretical remarks) can be found in Sites, Section 7.9.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #7779 by Reimundo Heluani on