  In this article, we apply the approach of relative algebraic geometry towards
analytic geometry to the category of bornological and Ind-Banach spaces
(non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne
dagger affinoid domains with their weak G-topology in this new language. We
prove an abstract recognition principle for the generators of their standard
topology (the morphisms appearing in the covers). We end with a sketch of an
emerging theory of dagger affinoid spaces over the integers, or any Banach
ring, where we can see the Archimedean and non-Archimedean worlds coming
together.
