Analytic geometry as relative algebraic geometry

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 notions of a Zariski toplogy. We show that one of these notions agrees with the G-topology 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 quasi-abelian categories of Banach spaces or modules as developed by Schneiders and Prosmans are very helpful. This is joint work with Kobi Kremnizer (Oxford).