Culf maps and edgewise subdivision
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915855225323520 |
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| author | Hackney, Philip Kock, Joachim |
| author_facet | Hackney, Philip Kock, Joachim |
| contents | We show that, for any simplicial space $X$, the $\infty$-category of culf maps over $X$ is equivalent to the $\infty$-category of right fibrations over $\operatorname{sd}(X)$, the edgewise subdivision of $X$. (When $X$ is a Rezk complete Segal or 2-Segal space, $\operatorname{sd}(X)$ is the twisted arrow category of $X$.) We give two proofs of independent interest; one exploiting comprehensive factorization and the natural transformation from the edgewise subdivision to the nerve of the category of elements, and another exploiting a new factorization system of ambifinal and culf maps, together with the right adjoint to edgewise subdivision. Using this main theorem, we show that the $\infty$-category of decomposition spaces and culf maps is locally an $\infty$-topos. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11191 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Culf maps and edgewise subdivision Hackney, Philip Kock, Joachim Algebraic Topology Category Theory 18N50, 55U10, 18N45, 18N60, 18N55, 18A32 We show that, for any simplicial space $X$, the $\infty$-category of culf maps over $X$ is equivalent to the $\infty$-category of right fibrations over $\operatorname{sd}(X)$, the edgewise subdivision of $X$. (When $X$ is a Rezk complete Segal or 2-Segal space, $\operatorname{sd}(X)$ is the twisted arrow category of $X$.) We give two proofs of independent interest; one exploiting comprehensive factorization and the natural transformation from the edgewise subdivision to the nerve of the category of elements, and another exploiting a new factorization system of ambifinal and culf maps, together with the right adjoint to edgewise subdivision. Using this main theorem, we show that the $\infty$-category of decomposition spaces and culf maps is locally an $\infty$-topos. |
| title | Culf maps and edgewise subdivision |
| topic | Algebraic Topology Category Theory 18N50, 55U10, 18N45, 18N60, 18N55, 18A32 |
| url | https://arxiv.org/abs/2210.11191 |