Eilenberg–MacLane Spectrum of F₁

• Eilenberg–MacLane Spectrum of F₁ is a construction that models the field with one element using repeated Segal delooping of discrete Γ-spaces to yield n-spheres.
• The method employs an iterative bar construction, where each n-sphere carries a free partial commutative monoid structure, linking combinatorial geometry with topology.
• Stable homotopy identification confirms that the spectrum recovers the classical sphere spectrum, highlighting rigidity in extension phenomena and discrete algebraic foundations.

The Eilenberg–MacLane spectrum of $\mathbb{F}_1$ provides a spectral homotopy-theoretic incarnation of the “field with one element,” constructed via repeated Segal delooping of a discrete $\Gamma$-space model of $\mathbb{F}_1$ (Beardsley, 2 Aug 2025). The resulting connective $\Omega$-spectrum consists of $n$-fold simplicial sets whose geometric realizations are $n$-spheres, each equipped with a canonical free partial commutative monoid structure (equivalently, the nerve of the free partial strict $n$-category with symmetric monoidal structure). This model reflects and generalizes the classical Eilenberg–MacLane spectrum construction for abelian groups to the context of algebra in characteristic one.

1. Discrete $\Gamma$-Spaces and $\mathbb{F}_1$-Modules

Discrete $\Gamma$-spaces are functors $X \colon \mathrm{Fin}_* \to \mathrm{Set}_*$, where $\mathrm{Fin}_*$ denotes the category of finite pointed sets. In the perspective of Connes–Consani, $\mathbb{F}_1$ may be realized as the inclusion functor $F \colon \mathrm{Fin}_* \to \mathrm{Set}_*$, treated as the “unit” object for the symmetric monoidal structure provided by Day convolution (Beardsley et al., 6 Apr 2024).

The category of $\mathbb{F}_1$-modules is thus identified with the category of pointed set-valued functors on $\mathrm{Fin}_*$, admitting a symmetric monoidal structure in which $F$ is the unit. The embedding of combinatorial geometry into $\mathbb{F}_1$-modules is achieved via the plasmic nerve functor $H$ and the functor $\Pi$ from the category of simple pointed matroids to plasmas (weakly unital, commutative hypermagmas), generalizing the classical Eilenberg–MacLane embedding.

2. Segal’s Delooping Functor and Spectrum Construction

Segal’s delooping functor $B$ applies to $\Gamma$-objects $X$ in pointed spaces or sets and formally produces a new $\Gamma$-object $BX$ capturing the “delooping” or iterated suspension of $X$. For “very special” $\Gamma$-spaces (such as those associated with abelian groups), repeated application of the delooping functor recovers the classical connective Eilenberg–MacLane spectrum.

For arbitrary discrete $\Gamma$-spaces—including those corresponding to $\mathbb{F}_1$—the functor is equally applicable but requires refined treatment when the Segal condition (which ensures equivalence between the spectrum and an infinite loop space) fails. The paper utilizes a distinct bar construction $\mathbb{B}$ for this context, iteratively building higher-dimensional simplicial objects: $$\mathbb{B}^n F(1) \cong \mathbb{S}^n := \square^n/\partial\square^n$$ where $\square^n$ is the free $n$-cube and $\partial\square^n$ its boundary. Each geometric realization yields the $n$-sphere, $S^n$.

3. Free Partial Commutative Monoid Structure and Partial Categories

Each $\mathbb{S}^n$ carries a free partial commutative monoid structure. The face maps of the $n$-fold simplicial set are determined by projections (“outer summands”) and fold maps (“merging adjacent faces”), encoding partial addition: $(x, y) \to x \vee y$ is only defined when one of $x$, $y$ is the basepoint.

The associated $\Gamma$-object $\mathbb{B}^n F$ is isomorphic to $(\mathbb{S}^n)^\vee$, the folding bar construction of the $n$-sphere, which can also be interpreted as the nerve of the free partial strict $n$-category with symmetric monoidal structure. In these categories, the tensor operation is partial—it is only defined if at least one argument is the unit element.

4. Geometric Realization and Stable Homotopy Identification

The geometric realization of the $n$-fold cube $\square^n$ is $[0,1]^n$, and the realization of the quotient by its boundary is $S^n$. This identifies the constituent spaces of the spectrum with the $n$-spheres, each equipped with the described algebraic structure. The symmetric monoidal structure on $\mathbb{F}_1$-modules ensures compatibility of these operations with the stable homotopy category.

The algebraic K-theory of $F$, $K(F)$, is equivalent to the sphere spectrum $S$ (Beardsley et al., 6 Apr 2024), confirming that stabilization via the plus-construction or Barratt–Priddy–Quillen theorem recovers the classical sphere spectrum from the $\mathbb{F}_1$-module context.

5. Extensions: Galois, Separable, and Étale Considerations

Analysis of Galois, separable, and étale extensions for Eilenberg–MacLane spectra (Betley, 2011) reveals a rigidity phenomenon: extensions of $HR$ in ring spectra are discrete if $R$ is concentrated in degree zero, and extra homotopical information does not produce new types of extensions in these cases. Specifically,

• Galois extensions: If $HR \to B$ satisfies the spectral Galois conditions, $B$ must be equivalent to an Eilenberg–MacLane spectrum $H(T_0(B))$ concentrated in degree zero.
• Separable extensions: Under restrictive algebraic hypotheses (notably no zero divisors in $T_0(B)$), separable extensions are again discrete.
• Étale extensions: For connective spectra, the étale condition forces the extension to be discrete, as any positive-dimensional homotopy group would detect nontrivial Hochschild or Kähler differential information.

For $H\mathbb{F}_1$, any “classical” extension must arise from discrete algebra, so that all “good” extensions are visible at the level of the ring $\mathbb{F}_1$. Nontrivial spectral phenomena would require nonconnective or “exotic” contributions, as in Mandell's examples.

6. Embeddings of Geometric and Algebraic Structures

The construction of the Eilenberg–MacLane spectrum of $\mathbb{F}_1$ interacts robustly with projective geometry and matroid theory. Projective geometries (encoded as simple pointed matroids with closure operators) embed fully faithfully into $\mathbb{F}_1$-modules via the plasmic nerve construction; this generalizes classical category embeddings and extends the reach of Segal’s nerve for commutative partial monoids (Beardsley et al., 6 Apr 2024). The explicit formulas

$$x_\kappa y = \begin{cases} \kappa(x, y) \setminus \{x, y, 0\}, & x \neq y \ \{x, 0\}, & x = y \end{cases}$$

capture combinatorial relations as algebraic operations, allowing the passage of closure data to the homotopy-theoretic context.

7. Further Directions and Open Problems

The results indicate that, within the context of the spectrum $H\mathbb{F}_1$, there is considerable rigidity: spectral constructions do not introduce new extensions absent from the discrete algebraic theory. Research directions include:

• Investigating the behavior of nonconnective étale extensions, which may present genuinely topological phenomena.
• Studying structured ring spectra and their extension theory to detect higher homotopy data beyond classical commutative rings.
• Applying these insights within motivic homotopy theory, K-theory, and “homotopy theory over $\mathbb{F}_1$.”

A plausible implication is that the Eilenberg–MacLane spectrum of $\mathbb{F}_1$, while embedding rich combinatorial and categorical data, recovers only the classical sphere spectrum and does not reveal “exotic” extensions unless the underlying objects are nonconnective or deviate from the discrete setup. This suggests stringent limitations on the potential for new spectral phenomena derived from $\mathbb{F}_1$, situating its homotopical role as foundational yet rigid within higher category theory and stable homotopy theory.