# Sites for étale $G$-spaces

This is a problem Sheaves in Geometry and Logic by MacLane and Moerdijk (problem 3.10)

Suppose we have a space $X$ on which a discrete group $G$ acts by homeomorphisms. Given an étale space $p:E \rightarrow X$, we call it an étale $G$-space if it has a $G$-action commuting with $p$. I'm to show these things form a Grothendieck topos by cooking up an explicit site. Now, the second part of the problem asks to show that if $G$ acts properly discontinuously on $X$, then these are actually just $Sh(X/G)$. I think this is fairly straightforward, since in this case quotients by $G$ and pullbacks along the map $X \rightarrow X/G$ will preserve "étaleness".

On the other extreme, I believe that if $G$ acts trivially on $X$ ($xg = x$ for all $x,g$), then an étale $G$-space is just an étale space with fiber-preserving homeomorphisms for each $g \in G$. It seems that, in this case, it seems like the given structure can be viewed as a functor from $G^{op}$ to $Sh(X)$, and a $G$-map between two étale $G$-sets will just be a natural transformation between two of these functors. The previous exercise shows how to make such categories into a sheaf on a site. In this case, the site will be $Open(X)\times G$, with covers of the form $S \times t_\star$, where $t_\star$ is the maximal sieve.

I'm just not sure how to get in between these two extremes! One commonality between these two things is that we seem to be building sheaves on a category whose objects are $G$-orbits of $X$.

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Your intuition is correct. Let us consider the category $\mathcal{C}$ whose objects are open subsets of $X$ and whose morphisms $U \to V$ are triples $(g, U, V)$, where $g \in G$ and $g \cdot U \subseteq V$. Notice that an isomorphism class in this category is a $G$-orbit in the frame of open subsets of $X$, and that the automorphism groups are the appropriate stabiliser subgroups.
What do presheaves on $\mathcal{C}$ look like? Well, to begin with, it is a presheaf on $X$ (since the category of open subsets of $X$ is a (non-full) subcategory of $\mathcal{C}$). Further, there is an "action" of $G$ on the sections that respects the action on $X$. So this begins to look like a good candidate for a site for $\mathbf{Sh}_G (X)$.
Now we must dream up a topology on $\mathcal{C}$. The easiest way to do this is to construct a functor $i : \mathcal{C} \to \mathbf{Sh}(X)$ and equip $\mathcal{C}$ with the largest topology that makes $i$ into a morphism of sites. Let $V$ be an object in $\mathcal{C}$, and define $i (V)$ to be the presheaf on $X$ given by $$i (V) (U) = \mathcal{C}(U, V)$$ with the evident restriction maps. It is not hard to see that this is a sheaf on $X$. Then define the covering families in $\mathcal{C}$ to be the ones that get sent to jointly epimorphic familes in $\mathbf{Sh} (X)$.
There are still several details to be verified – for instance, we should check that this actually defines a topology on $\mathcal{C}$, that $i : \mathcal{C} \to \mathbf{Sh} (X)$ so defined is a flat functor, and that the resulting category of sheaves is actually the right thing. I leave that to you – but I have to admit that I have not done the checking myself, so what I have said here might be completely wrong.