Dynkin Systems and the One-Point Geometry
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915213588037632 |
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| author | Beardsley, Jonathan |
| author_facet | Beardsley, Jonathan |
| contents | In this note I demonstrate that the collection of Dynkin systems on finite sets assembles into a Connes-Consani $\mathbb{F}_1$-module, with the collection of partitions of finite sets as a sub-module. The underlying simplicial set of this $\mathbb{F}_1$-module is shown to be isomorphic to the delooping of the Krasner hyperfield $\mathbb{K}$, where $1+1=\{0,1\}$. The face and degeneracy maps of the underlying simplicial set of the $\mathbb{F}_1$-module of partitions correspond to merging partition blocks and introducing singleton blocks, respectively. I also show that the $\mathbb{F}_1$-module of partitions cannot correspond to a set with a binary operation (even partially defined or multivalued) under the ``Eilenberg-MacLane'' embedding. These results imply that the $n$-fold sum of the Dynkin $\mathbb{F}_1$-module with itself is isomorphic to the $\mathbb{F}_1$-module of the discrete projective geometry on $n$ points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14769 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynkin Systems and the One-Point Geometry Beardsley, Jonathan Category Theory Algebraic Topology Combinatorics Probability 05A18, 18N50, 51E20, 60A10, 60A86, 16Y20, 18M80, 18A25, 14A23 In this note I demonstrate that the collection of Dynkin systems on finite sets assembles into a Connes-Consani $\mathbb{F}_1$-module, with the collection of partitions of finite sets as a sub-module. The underlying simplicial set of this $\mathbb{F}_1$-module is shown to be isomorphic to the delooping of the Krasner hyperfield $\mathbb{K}$, where $1+1=\{0,1\}$. The face and degeneracy maps of the underlying simplicial set of the $\mathbb{F}_1$-module of partitions correspond to merging partition blocks and introducing singleton blocks, respectively. I also show that the $\mathbb{F}_1$-module of partitions cannot correspond to a set with a binary operation (even partially defined or multivalued) under the ``Eilenberg-MacLane'' embedding. These results imply that the $n$-fold sum of the Dynkin $\mathbb{F}_1$-module with itself is isomorphic to the $\mathbb{F}_1$-module of the discrete projective geometry on $n$ points. |
| title | Dynkin Systems and the One-Point Geometry |
| topic | Category Theory Algebraic Topology Combinatorics Probability 05A18, 18N50, 51E20, 60A10, 60A86, 16Y20, 18M80, 18A25, 14A23 |
| url | https://arxiv.org/abs/2503.14769 |