The Eilenberg-MacLane Spectrum of \mathbb{F}_1 (2508.01524v1)

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.

• The paper presents a formal construction of the F1 spectrum using Segal’s delooping, linking Gamma-spaces with partial monoidal structures.
• It demonstrates that iterated deloopings yield geometric realizations homeomorphic to spheres and tori, enhancing insights into stable homotopy theory.
• The work embeds free partial commutative monoid structures into n-groupoids, opening avenues for extending spectral methods to multivalued operations.

The Eilenberg-MacLane Spectrum of $\mathbb{F}_1$: Formal Delooping and Partial Monoidal Structures

Introduction and Motivation

This paper investigates the construction of Eilenberg-MacLane spectra in the context of the "field with one element" ($\mathbb{F}_1$), leveraging the formalism of $\Gamma$-spaces and their deloopings. The work is motivated by the program initiated by Connes and Consani, which interprets $\mathbb{F}_1$-modules as pointed functors from finite pointed sets ($Fin_\ast$) to pointed sets, i.e., Segal $\Gamma$-sets. The central technical question addressed is: what is the spectrum associated to the base $\mathbb{F}_1$-module under Segal's delooping construction, and what algebraic and geometric structures does it encode?

Background: $\Gamma$-Spaces, Plasmas, and $\mathbb{F}_1$-Modules

The paper adopts the perspective that $\mathbb{F}_1$-modules are $\Gamma$-sets, i.e., pointed functors $Fin_\ast \to Set_\ast$, equipped with the Day convolution symmetric monoidal structure. The monoidal unit $F$ is the inclusion $Fin_\ast \hookrightarrow Set_\ast$. The notion of a commutative plasma is introduced as a generalization of commutative monoids, allowing for partial or multivalued addition. The initial partial commutative monoid $\mathbb{F}$, with $1+1 = \varnothing$, is identified as the algebraic structure underlying $F$.

A key technical result is the existence of a fully faithful right adjoint $\widehat{H}: \mathrm{CPlas} \hookrightarrow Mod_F$, which recovers the classical Eilenberg-MacLane embedding on Abelian groups and identifies $\widehat{H}\mathbb{F} \cong F$.

Segal's Delooping Functor and Iterated Bar Constructions

The delooping functor $\mathfrak{B}: Mod_F \to s_F$ is defined via pullback along a composite involving the smash product in $Fin_\ast$. This functor can be iterated, yielding $\mathfrak{B}^n: s^k Mod_F \to s^{k+n} Mod_F$. The bar construction $\mathtt{s}^\ast X$ for a $\Gamma$-object $X$ generalizes the classical bar construction for Abelian groups.

The paper emphasizes that, unlike the classical case, the geometric realization of the delooping functor for $F$ (which is not a very special $\Gamma$-space) is lossy, and thus the construction is performed at the level of $n$-fold simplicial sets.

Geometric Structure: Spheres and Tori as Deloopings

The main geometric result is that the $n$-fold delooping $\mathfrak{B}^n F\langle 1\rangle$ yields the $n$-fold simplicial set $\mathfrak{S}^n$, whose geometric realization is homeomorphic to the $n$-sphere $S^n$. More generally, the delooping of the corepresentable $\Gamma(n)$ yields the $n$-torus as a product of simplicial circles.

The construction is shown to produce $n$-fold simplicial sets with a single non-degenerate simplex in multidegree $[1,1,\ldots,1]$, and all other multisimplices degenerate or trivial. This reflects the minimal categorical and monoidal structure present in $F$.

Algebraic Structure: Free Partial Commutative Monoids and Strict $n$-Categories

Algebraically, $B^n F$ is identified as the functor $Fin_\ast \to s^n Set_\ast$ that encodes the free partial commutative monoid structure on the $n$-sphere. The partial monoid structure is realized via the fold map $S^n \vee S^n \to S^n$, defined only when one summand is the identity. This is formalized as the $\Gamma$-object $(\mathfrak{S}^n)^\vee$, the folding bar construction.

A key result is the existence of a monomorphism $$\iota_n: \mathfrak{B}^n F \hookrightarrow \mathfrak{K}(\mathbb{Z}/2, n)$$, embedding the partial monoid structure into the strict $n$-groupoid associated to $\mathbb{Z}/2$. The image corresponds to the "axes" in $(\mathbb{Z}/2)^d$ for $d$-simplices, reflecting the minimality of the structure.

The face maps in $B^n F$ are described explicitly: projections onto outer summands for $i=0,k$, and fold maps for $0 < i < k$. This matches the expected behavior for partial commutative monoids.

Implications and Future Directions

The construction provides a formal model for the Eilenberg-MacLane spectrum of $\mathbb{F}_1$, yielding a sequence of spaces (spheres) equipped with free partial commutative monoid structures. This model sits inside the classical infinite loop space machinery, but fails the Segal condition, reflecting the lack of full monoidal structure in $\mathbb{F}_1$-modules.

The results have implications for the paper of homotopy theory over $\mathbb{F}_1$, categorical models of partial algebraic structures, and the interface between stable homotopy theory and arithmetic geometry. The formalism suggests avenues for generalizing derived and spectral constructions to settings where only partial or multivalued operations are available.

Potential future developments include:

• Extending the formalism to more general classes of $F$-modules, such as projective geometries or hypergroups.
• Investigating the homotopical and categorical properties of spectra built from partial monoids.
• Exploring connections to motivic homotopy theory and absolute algebraic geometry.

Conclusion

This paper provides a rigorous computation of the Eilenberg-MacLane spectrum associated to the base $\mathbb{F}_1$-module, using Segal's delooping construction in the context of $\Gamma$-spaces. The resulting spectra are modeled by spheres equipped with free partial commutative monoid structures, reflecting the minimal algebraic content of $\mathbb{F}_1$. The work clarifies the formal relationship between partial algebraic structures and stable homotopy theory, and opens new directions for the paper of homotopy-theoretic phenomena in characteristic one.