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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2407.10891 |
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| _version_ | 1866911974385778688 |
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| author | Kolomatskaia, Astra |
| author_facet | Kolomatskaia, Astra |
| contents | We develop formulas that define permutahedral commutation coherence relations of all orders. To illustrate the result geometrically, we begin by defining a rigid transformation of the $(n+1)$-permutahedron into a $n$-cube of dimensions $1 \times 2 \times \cdots \times n$. With a fictitious assumption, we 'define' the corresponding coherence relations 'up to associativity' as an instance of a semi-simplicial type in the language of Displayed Type Theory. This is not a formal result in type theory, but we expect this to translate into one as soon as the problem of defining associahedral coherences is solved in a type theory with semi-simplicial types. On the other hand, this not-strictly-well-typed definition may be used to produce well-typed formulas in a restricted setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | You Wouldn't Permutahedron Kolomatskaia, Astra Category Theory Combinatorics We develop formulas that define permutahedral commutation coherence relations of all orders. To illustrate the result geometrically, we begin by defining a rigid transformation of the $(n+1)$-permutahedron into a $n$-cube of dimensions $1 \times 2 \times \cdots \times n$. With a fictitious assumption, we 'define' the corresponding coherence relations 'up to associativity' as an instance of a semi-simplicial type in the language of Displayed Type Theory. This is not a formal result in type theory, but we expect this to translate into one as soon as the problem of defining associahedral coherences is solved in a type theory with semi-simplicial types. On the other hand, this not-strictly-well-typed definition may be used to produce well-typed formulas in a restricted setting. |
| title | You Wouldn't Permutahedron |
| topic | Category Theory Combinatorics |
| url | https://arxiv.org/abs/2407.10891 |