The Eilenberg-MacLane Spectrum of \mathbb{F}_1
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911727636971520 |
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| author | Beardsley, Jonathan |
| author_facet | Beardsley, Jonathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01524 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Eilenberg-MacLane Spectrum of \mathbb{F}_1 Beardsley, Jonathan Algebraic Topology Category Theory 14A23, 08A55, 20N20, 18M05, 55P47, 57T30, 18N60, 55P42 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. |
| title | The Eilenberg-MacLane Spectrum of \mathbb{F}_1 |
| topic | Algebraic Topology Category Theory 14A23, 08A55, 20N20, 18M05, 55P47, 57T30, 18N60, 55P42 |
| url | https://arxiv.org/abs/2508.01524 |