Universal Properties of Variations of the Little Cubes Operads
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917383174619136 |
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| author | Arakawa, Kensuke |
| author_facet | Arakawa, Kensuke |
| contents | Given a map $B\to B\mathrm{Top}(n)$ of spaces, one can define a version $\mathbb{E}_{B}$ of the little cubes operad, whose construction is due to Lurie. We show that $\mathbb{E}_{B}$ enjoys the universal property that, for every $\infty$-operad $\mathcal{O}$, an operad map $\mathbb{E}_{B}\to\mathcal{O}$ is equivalent to a $\mathrm{Top}(n)$-equivariant map $B\times_{B\mathrm{Top}(n)}E\mathrm{Top}(n)\to\operatorname{Map}(\mathbb{E}_{n},\mathcal{O})$. This gives us an explicit diagram exhibiting $\mathbb{E}_{B}$ as a colimit of $\mathbb{E}_{n}$ parametrized by $B$. It also shows that locally constant factorization algebras satisfy descent, reproving a recent theorem of Matsuoka. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universal Properties of Variations of the Little Cubes Operads Arakawa, Kensuke Algebraic Topology 18M60, 18N60, 55U40 Given a map $B\to B\mathrm{Top}(n)$ of spaces, one can define a version $\mathbb{E}_{B}$ of the little cubes operad, whose construction is due to Lurie. We show that $\mathbb{E}_{B}$ enjoys the universal property that, for every $\infty$-operad $\mathcal{O}$, an operad map $\mathbb{E}_{B}\to\mathcal{O}$ is equivalent to a $\mathrm{Top}(n)$-equivariant map $B\times_{B\mathrm{Top}(n)}E\mathrm{Top}(n)\to\operatorname{Map}(\mathbb{E}_{n},\mathcal{O})$. This gives us an explicit diagram exhibiting $\mathbb{E}_{B}$ as a colimit of $\mathbb{E}_{n}$ parametrized by $B$. It also shows that locally constant factorization algebras satisfy descent, reproving a recent theorem of Matsuoka. |
| title | Universal Properties of Variations of the Little Cubes Operads |
| topic | Algebraic Topology 18M60, 18N60, 55U40 |
| url | https://arxiv.org/abs/2406.01084 |