Projective Geometries as $\mathbb{F}_1$-modules

We describe a fully faithful embedding of the category of axiomatic projective geometries into $\mathbb{F}_1$-modules, in the sense of Connes and Consani. In particular, the collineation group of a geometry is isomorphic to the group of automorphisms of its associated $\mathbb{F}_1$-module. Along th