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 category opposite to the category of commutative algebra objects in a closed symmetric monoidal category carries several interesting topologies introduced by Toen, Vaquie and Vezzosi. In the case that the symmetric monoidal category is the category of groups some of these topologies specialize to the ordinary Zariski topology on the category of affine schemes. We consider the case where the symmetric monoidal category is the category of Banach spaces over a non-Archimedean complete valuation field. We show in this case that one of the abstractly defined topologies specializes to the G- topology of Berkovich analytic geometry. 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). No background in derived algebraic geometry or non-Archimedean geometry will be assumed. This work is available at http://arxiv.org/abs/1312.0338.