Contributed talk: Smooth and proper categories in analytic geometry

In this talk, I will initially introduce the definition of smooth and proper category, and then explore how it can be used in algebraic and analytic geometry. This definition fits very well within the algebraic setting, as it allows for the characterization of smooth and proper algebraic varieties by examining their category of quasi-coherent sheaves. In the analytic context, such a characterization is more challenging to achieve; during the talk, I will aim to explain the difficulties and some results that we can obtain in this setting.