The “aﬃnoid” part of the theory I was trying to make certainly benefited
from discussions with Grothendieck, but where his help was essential was in how
to define a global rigid space by gluing aﬃnoids together. In that section of the
letters I say “we follow fully and faithfully a plan furnished by Grothendieck” and
to the best of my recollection that is not much of an exaggeration. The plan, with
its “h-structures” and “special coverings” is rather complicated and I’m not sure
I ever really understood it well. The definition of rigid space given in the present
book is certainly much simpler and more satisfying than the one Grothendieck and
I arrived at. That they are equivalent is far from obvious.
That’s the end of my involvement in the history. I regret that I am too ignorant
to say much about the further development of the theory. It was soon substantially
simplified by the introduction of rational subdomains by Gerritzen and Grauert.
Grothendieck’s vision of a grand unification quoted above did not take place, but
what happened was much more interesting, especially the great idea of Berkovich.
To understand how the theory has evolved, the present book is an excellent place