Cosmological Unstraightening
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913876540391424 |
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| author | Rasekh, Nima |
| author_facet | Rasekh, Nima |
| contents | The unstraightening construction due to Lurie establishes an equivalence between presheaves and fibrations, using one prominent model of $(\infty,1)$-categories, namely quasi-categories. In this work we generalize this result by proving that for all $\infty$-cosmoi of $(\infty,1)$-categories in the sense of Riehl and Verity, which includes quasi-categories but also complete Segal spaces or $1$-complicial sets, their corresponding notions of fibrations and presheaves are biequivalent $\infty$-cosmoi via a natural zig-zag of cosmological biequivalences. The major idea that makes this possible is a lift of the quasi-categorical unstraightening construction to a cosmological biequivalence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16342 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cosmological Unstraightening Rasekh, Nima Category Theory 18N60, 18N40, 18N50, 18N45 The unstraightening construction due to Lurie establishes an equivalence between presheaves and fibrations, using one prominent model of $(\infty,1)$-categories, namely quasi-categories. In this work we generalize this result by proving that for all $\infty$-cosmoi of $(\infty,1)$-categories in the sense of Riehl and Verity, which includes quasi-categories but also complete Segal spaces or $1$-complicial sets, their corresponding notions of fibrations and presheaves are biequivalent $\infty$-cosmoi via a natural zig-zag of cosmological biequivalences. The major idea that makes this possible is a lift of the quasi-categorical unstraightening construction to a cosmological biequivalence. |
| title | Cosmological Unstraightening |
| topic | Category Theory 18N60, 18N40, 18N50, 18N45 |
| url | https://arxiv.org/abs/2505.16342 |