The paper Étale cohomology of diamonds defines partial properness and compactification for maps between v-sheaves, and in particular for perfectoid spaces and rigid-analytic spaces. Recently when reading Lectures on Analytic Geometry I found that there may be similar concepts for analytic spaces. Here's my attempting definition (I still prefer the notation in Lectures on Condensed Mathematics):
An analytic ring map $\mathcal{A}\to\mathcal{B}$ is called proper if the analytic structure of the latter ring is induced from the former, i.e. $\mathcal{B}=\mathcal{A}\otimes_\underline{\mathcal{A}}\underline{\mathcal{B}}$. (Maybe one should ask for steadiness in addition.) This is clearly stable under base change. An analytic space map $f\colon X\to Y$ is called partially proper if for affine open $V\subseteq Y$, $f^{-1}(V)$ can be covered by affine opens proper over $V$, and proper if it is quasicompact, quasiseperated, and partially proper. (Maybe one should ask for seperatedness.) For proper $f$, it is easy to see that $f_*\colon\mathsf{D}(X)\to\mathsf{D}(Y)$ commutes with $\mathsf{CondAni}$-tensoring on both sides, so we can reasonably call it $f_!$ and take right adjoint $f^!$. For partially proper $f$ and affine $Y$, one can define $f_!$ as the left Kan extension of $f_*$ from the stable subcategory generated by $j_*\mathsf{D}(U)$ where $j\colon U\hookrightarrow X$ runs over the open affines proper over $Y$. This coincides with the definition in Masterclass in Condensed Mathematics Session 19&20 in case of Riemann surface, if I did not go wrong.
Define the compactification $\overline{\mathcal{B}}^{/\mathcal{A}}=\mathcal{A}\otimes_\underline{\mathcal{A}}\underline{\mathcal{B}}$. If $\underline{\mathcal{B}}\in\mathsf{D}(\mathcal{A})$ is nuclear, then this is steady over $\mathcal{A}$; if moreover $\mathcal{B}$ is steady over $\mathcal{A}$, then the canonical map $\overline{\mathcal{B}}^{/\mathcal{A}}\to\mathcal{B}$ is a steady localization, and one can prove that in this case the compactification of a steady localization is again a steady localization, so it can be globalized.
This is motivated by the discussion of discrete adic space in Lectures on Condensed Mathematics, and it covers both discrete adic space and complex-analytic geometry; but when trying to copy $j_!$ from there, a problem arises: In general there might be nothing like $\mathcal{B}_{\infty/\mathcal{A}}$ for steady localizations, even in the case the underlying condensed rings are the same; so $j^*$ lacks a left adjoint.
So the questions are:
How generally are this "properness" and the six-functor formalism reasonable, or we have to work case by case in several kinds of geometry? I think it should also work for rigid-analytic geometry.
Perfectoid spaces and rigid-analytic spaces can be made analytic spaces. Does this "properness" coincide with that of diamonds?
Is it reasonable to talk about étale sheaves on analytic spaces, or the analytic formalism is way too general?
