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

Given a very special $\mathrm{\Gamma }$-space $X$, repeated application of Segal’s delooping functor produces the constituent spaces of the associated connective $\mathrm{\Omega }$-spectrum. In particular, by applying this construction to $\mathit{\text{discrete}}$ very special $\mathrm{\Gamma }$-spaces (a.k.a.~Abelian groups), one recovers Eilenberg-MacLane spectra. The delooping functor is entirely formal, however, and can be applied to arbitrary $\mathrm{\Gamma }$-spaces without any conditions. Work of Connes and Consani suggests that the ``field with one element’’ can be fruitfully realized as a (discrete) $\mathrm{\Gamma }$-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 $\mathit{\text{free}}\mathit{\text{partial}}\mathit{\text{commutative}}\mathit{\text{monoid}}$ structures. Equivalently, they are the (nerves of the) free partial strict $n$-categories with free partial symmetric monoidal structures.