The category of necklaces is Reedy monoidal
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914731234689024 |
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| author | Marques, Violeta Borges Mertens, Arne |
| author_facet | Marques, Violeta Borges Mertens, Arne |
| contents | In the first part of this note we further the study of the interactions between Reedy and monoidal structures on a small category, building upon the work of Barwick. We define a Reedy monoidal category as a Reedy category $\mathcal{R}$ which is monoidal such that for all symmetric monoidal model categories $\textbf{A}$, the category $\mathrm{Fun}\left(\mathcal{R}^{\mathrm{op}}, \textbf{A}\right)_{\mathrm{Reedy}}$ is model monoidal when equipped with the Day convolution. In the second part, we study the category $\mathcal{N}ec$ of necklaces, as defined by Baues and Dugger-Spivak. Making use of the combinatorial description present in arXiv:2302.02484v1, we streamline some proofs from the literature, and finally show that $\mathcal{N}ec$ is simple Reedy monoidal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12796 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The category of necklaces is Reedy monoidal Marques, Violeta Borges Mertens, Arne Category Theory Algebraic Topology 18M05, 18N40 (Primary), 05E45 (Secondary) In the first part of this note we further the study of the interactions between Reedy and monoidal structures on a small category, building upon the work of Barwick. We define a Reedy monoidal category as a Reedy category $\mathcal{R}$ which is monoidal such that for all symmetric monoidal model categories $\textbf{A}$, the category $\mathrm{Fun}\left(\mathcal{R}^{\mathrm{op}}, \textbf{A}\right)_{\mathrm{Reedy}}$ is model monoidal when equipped with the Day convolution. In the second part, we study the category $\mathcal{N}ec$ of necklaces, as defined by Baues and Dugger-Spivak. Making use of the combinatorial description present in arXiv:2302.02484v1, we streamline some proofs from the literature, and finally show that $\mathcal{N}ec$ is simple Reedy monoidal. |
| title | The category of necklaces is Reedy monoidal |
| topic | Category Theory Algebraic Topology 18M05, 18N40 (Primary), 05E45 (Secondary) |
| url | https://arxiv.org/abs/2308.12796 |