higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The genus of a surface is an integer that helps classify the surface’s homeomorphism class. It gives the number of handles or ‘cross caps’ that are in a sum decomposition of the surface.
Various approaches to the classification of surfaces are described in introductory texts such as