Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913304614535168 |
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| author | Beardsley, Jonathan Nakamura, So |
| author_facet | Beardsley, Jonathan Nakamura, So |
| contents | We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\fun$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some time elaborating its structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules Beardsley, Jonathan Nakamura, So Category Theory Combinatorics 51D20, 51D10, 20N20, 14A23, 18A25, 18B10, 18C40 We describe a fully faithful embedding of projective geometries, given in terms of closure operators, into $\mathbb{F}_1$-modules, in the sense of Connes and Consani. This factors through a faithful functor out of simple pointed matroids. This follows from our construction of a fully faithful embedding of weakly unital, commutative hypermagmas into $\fun$-modules. This embedding is of independent interest as it generalizes the classical Eilenberg-MacLane embedding for commutative monoids and recovers Segal's nerve construction for commutative partial monoids. For this reason, we spend some time elaborating its structure. |
| title | Projective Geometries and Simple Pointed Matroids as $\mathbb{F}_1$-modules |
| topic | Category Theory Combinatorics 51D20, 51D10, 20N20, 14A23, 18A25, 18B10, 18C40 |
| url | https://arxiv.org/abs/2404.04730 |