In this talk I will explain a category theoretic perspective on geometry. Starting with a category of local objects (of and algebraic nature), and a (Grothendieck)
topology on it, one can define global objects such as schemes and stacks. Examples of this approach are algebraic, analytic, differential geometries and also more exotic geometries such as analytic and differential geometry over the integers and analytic geometry over the field with one element. In this approach the notion of a point is not primary but is derived from the local to global structure. The Zariski and Huber spectra are recovered in this way, and we also get new spectra which might be of interest in model theory.
- Logic Seminar