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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2501.00416 |
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| _version_ | 1866917882101760000 |
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| author | Willerton, Simon |
| author_facet | Willerton, Simon |
| contents | This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a number -- the distance -- such that the triangle inequality is satisfied; a small category is a set of objects such that between each pair of objects there is a set -- the hom-set -- such that elements of the hom-sets can be composed. The analogy between the structures that can be made in to a common generalization of the two structures, so that both are examples of enriched categories. This gives a bridge between category theory and metric space theory. I will describe this and three examples from around mathematics where this perspective has been useful or interesting. The examples are related to the tight span, the magnitude and the Legendre-Fenchel transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric-like spaces as enriched categories: three vignettes Willerton, Simon Category Theory This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a number -- the distance -- such that the triangle inequality is satisfied; a small category is a set of objects such that between each pair of objects there is a set -- the hom-set -- such that elements of the hom-sets can be composed. The analogy between the structures that can be made in to a common generalization of the two structures, so that both are examples of enriched categories. This gives a bridge between category theory and metric space theory. I will describe this and three examples from around mathematics where this perspective has been useful or interesting. The examples are related to the tight span, the magnitude and the Legendre-Fenchel transform. |
| title | Metric-like spaces as enriched categories: three vignettes |
| topic | Category Theory |
| url | https://arxiv.org/abs/2501.00416 |