Invertibility and parity in symmetric monoidal categories
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914480230760448 |
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| author | Gurski, Nick Johnson, Niles |
| author_facet | Gurski, Nick Johnson, Niles |
| contents | We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers $\pm$1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15142 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Invertibility and parity in symmetric monoidal categories Gurski, Nick Johnson, Niles Category Theory Algebraic Topology We introduce a notion of parity for formal morphisms between invertible objects and use it to prove a corresponding coherence theorem. Parity is conceptually similar to the sign of underlying permutations, but not defined as such. To give complete details, this work includes a thorough treatment of the free permutative category on an invertible generator, its skeletal model, known as the super integers, and an equivalence between them classified by the pair of integers $\pm$1. Our approach is organized and clarified as an application of 2-monadic algebra, particularly the concept of flexibility and the Lack model structure. The final section contains a number of examples applying the main results. |
| title | Invertibility and parity in symmetric monoidal categories |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2604.15142 |