Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $X$.

Answers are nice, but information about how to think about these things is preferred!

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.