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

Abstract

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 the way, we describe a version of the classifying space construction for a broad generalization of Abelian groups which we call plasmas. Our construction subsumes Segal's existing delooping constructions for Abelian groups, commutative monoids and partial commutative monoids.

v2: removed several minor errors that did not affect the main results, refined exposition

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