I've been reading about adic spaces, and I couldn't help but wonder what would happen to the theory if one included in the definition of $Spa$ Archimedean valuations as well...?

Is there a weird Arakelov-type cousin of adic spaces? Would there be any merit for such a thing?

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    $\begingroup$ I guess the first question is: what is an Archimedean valuation? (with general value group) $\endgroup$ Aug 30 '18 at 8:34
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    $\begingroup$ You probably need to change the definition of affinoid ring $(A,A^+)$ so that $A^+$ is not a subring, but a multiplicative submonoid. Otherwise, $\operatorname{Spa}(\mathbb{Z})$ lacks an archimedean branch. $\endgroup$
    – S. Carnahan
    Aug 30 '18 at 10:37

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