Modular operads, iterated distributive laws and a nerve theorem for circuit algebras
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918385091084288 |
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| author | Raynor, Sophie |
| author_facet | Raynor, Sophie |
| contents | Circuit algebras are a symmetric version of Jones's planar algebras. They originated in quantum topology as a framework for encoding virtual crossings. This paper extends existing results for modular operads to construct a graphical calculus and monad for general circuit algebras and prove an abstract nerve theorem. The proof relies on a subtle interplay between distributive laws and abstract nerve theory, and provides extra insights into the underlying structures. Oriented circuit algebras are equivalent to wheeled props and specialisations of the results to wheeled props follow as straightforward corollaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20262 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modular operads, iterated distributive laws and a nerve theorem for circuit algebras Raynor, Sophie Category Theory Algebraic Topology Quantum Algebra 18M85 (Primary) 18C15, 57K16 (Secondary) Circuit algebras are a symmetric version of Jones's planar algebras. They originated in quantum topology as a framework for encoding virtual crossings. This paper extends existing results for modular operads to construct a graphical calculus and monad for general circuit algebras and prove an abstract nerve theorem. The proof relies on a subtle interplay between distributive laws and abstract nerve theory, and provides extra insights into the underlying structures. Oriented circuit algebras are equivalent to wheeled props and specialisations of the results to wheeled props follow as straightforward corollaries. |
| title | Modular operads, iterated distributive laws and a nerve theorem for circuit algebras |
| topic | Category Theory Algebraic Topology Quantum Algebra 18M85 (Primary) 18C15, 57K16 (Secondary) |
| url | https://arxiv.org/abs/2412.20262 |