I will review symmetric monoidal categories and explain how one
can work with "algebras and modules" in such a category. Toen, Vaquie, and
Vezzosi promoted the study of algebraic geometry relative to a closed
symmetric monoidal category. By considering the closed symmetric monoidal
category of Banach spaces, we recover various aspects of Berkovich
analytic
geometry. The opposite category to commutative algebra objects in a closed
symmetric monoidal category has a few different notion of a Zariski
toplogy. We show that one of these notions agrees with the Gtopology of
Berkovich theory and embed Berkovich analytic geometry into
these abstract versions of algebraic geometry. We will describe the basic
open sets in this topology and what algebras they correspond to. These
algebras play the same role as the basic localizations which you get from
a ring by inverting a single element. In our context, the quasiabelian
categories of Banach spaces or modules as developed by Schneiders and
Prosmans are very helpful. This is joint work with Kobi Kremnizer
(Oxford).
