Categorification and Geometry

The key principle in Grothendieck's algebraic geometry is that every commutative ring be considered as the ring of functions on some geometric object. Clausen and Scholze have introduced a categorification of algebraic and analytic geometry, where the key principle is that every stable closed symmetric monoidal infinity-category be considered as the infinity-category of quasi-coherent modules on some geometric object. In this talk, I will expland on this principle as well as Scholze's philosophy that *every* cohomology theory should arise from this picture, complete with a six-functor formalism of categories of coefficients. The Hahn-Raksit-Wilson even filtration and Efimov continuity are key ingredients.