The Eilenberg-MacLane Spectrum of \mathbb{F}_1

Abstract

Given a very special $Γ$-space $X$, repeated application of Segal's delooping functor produces the constituent spaces of the associated connective $Ω$-spectrum. In particular, by applying this construction to \textit{discrete} very special $Γ$-spaces (a.k.a.~Abelian groups), one recovers Eilenberg-MacLane spectra. The delooping functor is entirely formal, however, and can be applied to arbitrary $Γ$-spaces without any conditions. Work of Connes and Consani suggests that the ``field with one element'' can be fruitfully realized as a (discrete) $Γ$-space (which localizes to the classical sphere spectrum). This note computes Segal's deloopings of this model of $\mathbb{F}_1$. They are $n$-fold simplicial sets whose geometric realizations are the $n$-spheres, equipped with \textit{free partial commutative monoid} structures. Equivalently, they are the (nerves of the) free partial strict $n$-categories with free partial symmetric monoidal structures.