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This question is motivated by the following example : let $X$ be a variety over a field $k$, with algebraic closure $\bar{k}$. The Galois group $G_k:=\mathrm{Gal}(\bar{k}/k)$ acts on $X_{\bar{k}}:=X\times_k \bar{k}$ via the second factor, and for the projection $p:X_{\bar{k}}\to X$ and an étale sheaf $F$ on $X_{ét}$, the sheaf $p^\ast F$ on $X_{\bar{k}}$ has a $G_k$-equivariant structure and we have $\Gamma_X(F)=\Gamma(G_K,\Gamma_{X_{\bar{k}}}(p^\ast F))$. Actually we have more : if $F$ and $G$ are bounded complexes of étale sheaves of abelian groups on $X$, then $$R\Gamma(G_k,R\mathrm{Hom}_{X_{\bar{k}}}(p^\ast F,p^\ast G))=R\mathrm{Hom}_X(F,G)$$

I'm interested in potential generalizations of this, but I don't know very much about equivariant sheaves and toposes so hopefully this will not be too naive. Here are my questions:

  1. What does it mean to be a $G_k$-equivariant sheaf in the above ? I've heard only about $G$-equivariant sheaves for $G$ a discrete group, but here $G_k$ is profinite.

  2. Do we actually have an equivalence of categories between étale sheaves on $X$ and $G_k$-equivariant étale sheaves on $X_{\bar{k}}$ ?

  3. Is there some sense in which $X=X_{\bar{k}}/G_k$ ?

  4. Does it make sense to say that the étale topos of $X_{\bar{k}}$ is the "universal $G_k$-topos" over the étale topos of $X$ ? Is there a way to give a precise meaning to that ?

  5. In general, if $\cal{T}$ is a topos and $G$ is a profinite group, does there exist a "universal $G$-topos" over $\cal{T}$ ? By that I mean a topos $\hat{\cal{T}}$ with an "action" of $G$ and a map $\pi:\hat{\cal{T}}\to \cal{T}$ such that $\Gamma_{\cal{T}}(F)=\Gamma(G,\Gamma_{\hat{\cal{T}}}(\pi^\ast F))$ for $F\in\cal{T}$ and $$R\Gamma(G,R\mathrm{Hom}_{\bar{\cal{T}}}(p^\ast F,p^\ast G))=R\mathrm{Hom}_{\cal{T}}(F,G)$$ for $F$ and $G$ in the "bounded derived category of abelian group objects" ?

  6. Specifically, does it exist for $\cal{T}=\mathrm{Sh}((\mathrm{Spec}(\mathcal{O}_K))_{ét})$ the (small) étale topos of the ring of integers $\mathcal{O}_K$ in a global or local field $K$ and $G=\mathrm{Gal}(\bar{K}/K)$ ? Is that universal topos given by sheaves on a familiar site ?

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  • $\begingroup$ For the two first questions I found a reference in SGA7, XIII, 1.1. 3. Point 2 is true and the action of $G_k$ on a sheaf $F$ on $X_{\bar{k}}$ is said to be continuous if for a quasicompact $U$ étale over $X$, the sections $F(U_{\bar{k}})$ form a discrete $G_k$-module. $\endgroup$ Nov 3, 2021 at 17:51

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Suppose that $\mathcal{T}$ is a topos, and let $G$ be a topological group. If you want there to be a "universal $G$-topos" over $\mathcal{T}$, then you need a geometric morphism $f : \mathcal{T} \to \mathbf{Cont}(G)$, where $\mathbf{Cont}(G)$ is the topos of sets with a continuous $G$-action. There is also a geometric morphism $p : \mathbf{Sets} \to \mathbf{Cont}(G)$, with $p^*$ the forgetful functor. The universal $G$-topos over $\mathcal{T}$ is then the pullback of $p$ along $f$.

For example, if $\mathcal{T}=\mathbf{Sh}(S^1)$, then there is a geometric morphism $f : \mathbf{Sh}(S^1) \to \mathbf{Cont}(\mathbb{Z})$, where $\mathbb{Z}$ is the discrete group of integers under addition. Here $f^*$ sends the $\mathbb{Z}$-set $\mathbb{Z}$ to the sheaf corresponding to the projection $\mathbb{R} \to S^1, t \mapsto e^{it}$ (this completely determines $f^*$ because $f^*$ preserves colimits). If you then compute the pullback of $p : \mathbf{Sets} \to \mathbf{Cont}(\mathbb{Z})$ along $f$, you get the "universal $\mathbb{Z}$-topos" over $\mathbf{Sh}(S^1)$, which is given by $\mathbf{Sh}(\mathbb{R})$.

In your setting, if $X$ is a variety over a field $k$, then the morphism of schemes $X \to \mathrm{Spec}(k)$ induces a geometric morphism between the small étale toposes $X_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$. Further you can prove that $\mathrm{Spec}(k)_\mathrm{\acute{e}t} \simeq \mathbf{Cont}(G_k)$, where $G_k$ is the absolute Galois group of $k$ (with its usual topology).

So in this case we do have a geometric morphism $X_\mathrm{\acute{e}t} \to \mathbf{Cont}(G_k)$, so it makes sense to talk about the universal $G_k$-topos over $X_\mathrm{\acute{e}t}$.

I don't know precisely how to prove that the universal $G_k$-topos over $X_\mathrm{\acute{e}t}$ is equivalent to $(X_{\bar{k}})_\mathrm{\acute{e}t}$. There are two strategies:

  1. The point $p : \mathbf{Sets} \to \mathbf{Cont}(G_k)$ agrees with the natural geometric morphism $\mathrm{Spec}(\bar{k})_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$. So if you show that this pseudopullback of small étale toposes (in the category of toposes) is computed by taking the pullback of the schemes, then this finishes the proof. EDIT: in this MathOverflow question it is claimed that this holds in the relevant case, because $\mathrm{Spec}(\bar{k})$ is qcqs and pro-étale over $\mathrm{Spec}(k)$.
  2. If $X$ is a topological space with an action of a discrete group $G$, then the universal $G$-topos over the topos of $G$-equivariant sheaves $\mathbf{Sh}_G(X)$ is given by $\mathbf{Sh}(X)$. Maybe one can show the following more general statement: that if $G$ is a topological group acting continuously on a topos $\mathcal{E}$, then the universal $G$-topos over $\mathbf{Sh}_G(\mathcal{E})$ is given by $\mathcal{E}$. I don't know how to make these defintions precise though.

EDIT: Here is a "topos-theoretic proof" of the property $\Gamma_X(F) = \Gamma(G_K,\Gamma_{X_\bar{k}}(q^*F))$ that you mentioned. I use here the name $q$ for the projection $(X_\bar{k})_\mathrm{\acute{e}t} \to X_\mathrm{\acute{e}t}$. I'm not sure if this will be helpful to you, but I'll add it for future reference.

Consider the geometric morphisms $p : \mathrm{Spec}(\bar{k})_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$ and $f : X_\mathrm{\acute{e}t} \to \mathrm{Spec}(k)_\mathrm{\acute{e}t}$. The pullback of $p$ along $f$ is $q$ as above, and I'll write $g$ for the pullback of $f$ along $p$.

I claim $f$ is a tidy geometric morphism. Because $p$ is an open surjection, it is enough to show that $g$ is tidy (Johnstone's Elephant, C.5.1.7). I added a proof that $g$ is tidy here. So $f$ is tidy as well.

Since $f$ is tidy, the Beck—Chevalley condition $p^*f_* \simeq g_*q^*$ holds (Johnstone's Elephant, C.3.4.11). Applying this to a sheaf $F$ gives $p^*f_*F \simeq g_*(q^*F) = \Gamma_{X_\mathrm{\acute{e}t}}(q^*F)$. This means that $f_*F$ has as underlying set precisely $\Gamma_{X_\mathrm{\acute{e}t}}(q^*F)$, and then there is a certain $G_k$-action on it. Taking the fixed points under the $G_k$-action amounts to taking global sections $\Gamma_k$ of the sheaf. So we get: $\Gamma(G_K,\Gamma_{X_\bar{k}}(q^*F)) = \Gamma_{k}(f_*F) = \Gamma_X(F).$

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  • $\begingroup$ This is awesome ! I have a few questions and remarks. 1) By pullback of topoi do you mean the pullback in the 2-category of topoi I presume ? 2) Can we compute an explicit presenting site for the pullback ? 3) I'm not sure I understand what a G-equivariant sheaf is with your definition, do you have some reference I could read ? $\endgroup$ Nov 4, 2021 at 12:43
  • $\begingroup$ Yes, it is indeed the pseudopullback in the 2-category of topoi. If the two morphisms defining the pullback come from morphisms of sites between sites that have finite limits (I believe this is the case in your setting), then you can compute a presenting site for the pullback as well, see here. I don't know how practical this calculation is in your case. $\endgroup$ Nov 4, 2021 at 13:11
  • $\begingroup$ About $G$-equivariant sheaves on topological spaces: there is an equivalence of categories between sheaves on $X$ and local homeomorphisms $Y \to X$. A $G$-equivariant sheaf then corresponds by definition to a local homeomorphism $\pi : Y \to X$ together with a continuous $G$-action on $Y$ such that $\pi(g \cdot x) = g \cdot \pi(x)$. The category of $G$-equivariant sheaves is then a Grothendieck topos. If $G$ is discrete, then an explicit site of definition is given in Johnstone's Elephant, Example 2.1.11(c), p. 76. $\endgroup$ Nov 4, 2021 at 13:16

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