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

TL;DR

This paper explores the construction of the Eilenberg-MacLane spectrum for the field with one element () using Segal's delooping functor applied to a -model as a -space, revealing new geometric and categorical structures.

Contribution

It computes Segal's deloopings of the -model, showing they form n-fold simplicial sets with geometric realizations as n-spheres, linking -geometry to higher categorical structures.

Findings

Deloopings are n-fold simplicial sets with n-sphere realizations

The structures are free partial commutative monoids

Equivalent to nerves of free partial strict n-categories

Abstract

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