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I will present a 'categorical' way of doing analytic geometry in which analytic geometry is seen as a precise analogue of algebraic geometry.
Our approach works for both complex analytic geometry and p-adic analytic geometry in a uniform way. I will focus on the idea of an 'open set' as used in various geometrical theories and how it is characterized
categorically. In order to do this, we need to study algebras and their modules in the category of Banach spaces. The categorical characterization that we need uses homological algebra in these 'quasi-abelian' categories which is work of Schneiders and Prosmans. In fact, we work with the larger category of Ind-Banach spaces for reasons I will explain. This gives us a way to establish foundations of derived analytic geometry (my joint project with Kobi Kremnizer). We compare this approach with standard standard notions such as the theory of affinoid algebras, Grosse-Klonne's theory of dagger algebras (over-convergent functions), the theory of Stein domains and others. I will formulate derived analytic geometry following the relative algebraic geometry approach of Toen, Vaquie and Vezzosi.

This talk involves various joint work with Federico Bambozzi and Kobi Kremnizer.