Projective Geometries and Simple Pointed Matroids as ${\mathbb{F}}_1$-modules

We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into ${\mathbb{F}}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\text{fun}$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal’s nerve construction for commutative partial monoids. For this reason, we spend some time elaborating its structure.