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Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, as well as for each adissible open subset $U \subset X$ a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible open subsets $U_i \subset X$, called admissible coverings, in such a way that the following axioms are satisfied:

Questions: Can the locale $X_{\mathrm{loc}}$ be described more explicitly? Is it spacial? Is there a topology on the set $X$ such that $X_{\mathrm{loc}}$ is given by the open subsets of this topology? (I thik that this is not in general the case.) Is there a topological space $X'$ satisfying $\mathrm{Sh}(X') = \mathrm{Sh}(X)$ and a G-topology on the underlying set of $X'$ together with a natural map $\beta \colon X \to X'$ of G-topological spaces, such that the operation $\beta^{-1}$ identifies the admissible open subsets of $X'$ with those of $X$ and the same holds for admissible coverings? I think that this is the case in the second example below.

Edit: Here is another remark: Any admssible open subset $U \hookrightarrow X$ gives rise to a morphism of topoi $\mathrm{Sh}(U) \to \mathrm{Sh}(X)$ and hence (by [MLM92, Prop. 2 in Sec. IX.5]) to a morphism $U_{\mathrm{loc}} \to X_{\mathrm{loc}}$. I think that by [MLM92, Prop. 5 (ii) in Sec. IX.5] this map is an embedding of locales, so both the ordered set of admissible open subsets of $X$ and the frame of opens of $X_{\mathrm{loc}}$ should embed in the frame of sublocales of $X_{\mathrm{loc}}$. Can we describe the relation between the two within this bigger poset?

Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, as well as for each admissible open subset $U \subset X$ a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible open subsets $U_i \subset X$, called admissible coverings, in such a way that the following axioms are satisfied:

Questions: Can the locale $X_{\mathrm{loc}}$ be described more explicitly? Is it spacial? Is there a topology on the set $X$ such that $X_{\mathrm{loc}}$ is given by the open subsets of this topology? (I think that this is not in general the case.) Is there a topological space $X'$ satisfying $\mathrm{Sh}(X') = \mathrm{Sh}(X)$ and a G-topology on the underlying set of $X'$ together with a natural map $\beta \colon X \to X'$ of G-topological spaces, such that the operation $\beta^{-1}$ identifies the admissible open subsets of $X'$ with those of $X$ and the same holds for admissible coverings? I think that this is the case in the second example below.

Edit: Here is another remark: Any admissible open subset $U \hookrightarrow X$ gives rise to a morphism of topoi $\mathrm{Sh}(U) \to \mathrm{Sh}(X)$ and hence (by [MLM92, Prop. 2 in Sec. IX.5]) to a morphism $U_{\mathrm{loc}} \to X_{\mathrm{loc}}$. I think that by [MLM92, Prop. 5 (ii) in Sec. IX.5] this map is an embedding of locales, so both the ordered set of admissible open subsets of $X$ and the frame of opens of $X_{\mathrm{loc}}$ should embed in the frame of sublocales of $X_{\mathrm{loc}}$. Can we describe the relation between the two within this bigger poset?

Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, as well as, for each adissible open subset $U \subset X$, a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible open subsets $U_i \subset X$, called admissible coverings, in such a way that the following axioms are satisfied:

Questions: Can the locale $X_{\mathrm{loc}}$ be described more explicitly? Is it spacial? Is there a topology on the set $X$ such that $X_{\mathrm{loc}}$ is given by the open subsets of this topology? (I thik that this is not in general the case.) Is there a topological space $X'$ satisfying $\mathrm{Sh}(X') = \mathrm{Sh}(X)$ and a G-topology on the underlying set of $X'$ together with a natural map $\beta \colon X \to X'$ of G-topological spaces, such that the operation $\beta^{-1}$ identifies the admissible open subsets of $X'$ with those of $X$ and the same holds for admissible coverings? I think that this is the case in the second example below.

Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, as well as for each adissible open subset $U \subset X$ a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible open subsets $U_i \subset X$, called admissible coverings, in such a way that the following axioms are satisfied:

Questions: Can the locale $X_{\mathrm{loc}}$ be described more explicitly? Is it spacial? Is there a topology on the set $X$ such that $X_{\mathrm{loc}}$ is given by the open subsets of this topology? (I thik that this is not in general the case.) Is there a topological space $X'$ satisfying $\mathrm{Sh}(X') = \mathrm{Sh}(X)$ and a G-topology on the underlying set of $X'$ together with a natural map $\beta \colon X \to X'$ of G-topological spaces, such that the operation $\beta^{-1}$ identifies the admissible open subsets of $X'$ with those of $X$ and the same holds for admissible coverings? I think that this is the case in the second example below.

Edit: Here is another remark: Any admssible open subset $U \hookrightarrow X$ gives rise to a morphism of topoi $\mathrm{Sh}(U) \to \mathrm{Sh}(X)$ and hence (by [MLM92, Prop. 2 in Sec. IX.5]) to a morphism $U_{\mathrm{loc}} \to X_{\mathrm{loc}}$. I think that by [MLM92, Prop. 5 (ii) in Sec. IX.5] this map is an embedding of locales, so both the ordered set of admissible open subsets of $X$ and the frame of opens of $X_{\mathrm{loc}}$ should embed in the frame of sublocales of $X_{\mathrm{loc}}$. Can we describe the relation between the two within this bigger poset?

Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, as well as, for each adissible open subset $U \subset X$, a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible opensubsets $U_i \subset X$, called admissible coverings, in such a way that the following axioms are satisfied:

In this case the category of sheaves agrees (I think) with the category of sheaves on the ordinary topological space $\mathbb{R}$ with its Euclidean topology. If necessary, I can provide a proof.

For another example (which is actually quite similar), let $\mathrm{Sp}(A)$ be the rigid analytic space associated to a $K$-affinoid algebra $A$ where $K$ is a non-archimedean field. It is a G-topological space whose points are the maximal ideals of $A$, whose admissible open subsets are the affinoid domains and whose admissible coverings are the finite ones. Then the sheaves on $\mathrm{Sp}(A)$ agree with sheaves on $\mathscr{M}(A)$, the Berkovich space associated to $A$ with its Berkovich topology. There is a canonical map $\mathrm{Sp}(A) \to \mathscr{M}(A)$ identifying the affinoid domains of $\mathrm{Sp}(A)$ with the affinoid domains of $\mathscr{M}(A)$ in the Berkovich sense.

Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, as well as, for each adissible open subset $U \subset X$, a system of certain distinguished set-theoretic coverings $U = \bigcup_{i \in I} U_i$ by other admissible open subsets $U_i \subset X$, called admissible coverings, in such a way that the following axioms are satisfied:

In this case the category of sheaves agrees (I think) with the category of sheaves on the ordinary topological space $\mathbb{R}$ with its Euclidean topology. If necessary, I can provide a proof.

For another example (which is actually quite similar), let $\mathrm{Sp}(A)$ be the rigid analytic space associated to a $K$-affinoid algebra $A$ where $K$ is a non-archimedean field. It is a G-topological space whose points are the maximal ideals of $A$, whose admissible open subsets are the affinoid domains and whose admissible coverings are the finite ones. Then the sheaves on $\mathrm{Sp}(A)$ agree with sheaves on $\mathscr{M}(A)$, the Berkovich space associated to $A$ with its Berkovich topology. There is a canonical map $\mathrm{Sp}(A) \to \mathscr{M}(A)$ identifying the affinoid domains of $\mathrm{Sp}(A)$ with the affinoid domains of $\mathscr{M}(A)$ in the Berkovich sense. As remarked by Piotr Achinger below, the Berkovich spectrum should be replaced with the Huber spectrum here.