A model structure for Grothendieck fibrations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909185694760960 |
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| author | Moser, Lyne Sarazola, Maru |
| author_facet | Moser, Lyne Sarazola, Maru |
| contents | We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category $\mathcal{C}$. For the discrete case, we build a model structure on the slice $\mathrm{Cat}_{/\mathcal{C}}$, Quillen equivalent to the projective model structure on $[\mathcal{C}^{\mathrm{op}},\mathrm{Set}]$ via the classical category of elements construction. The cartesian case requires the use of markings, and we define a model structure on the slice $\mathrm{Cat}^+_{/\mathcal{C}}$, Quillen equivalent to the projective model structure on $[\mathcal{C}^{\mathrm{op}},\mathrm{Cat}]$ via a marked version of the Grothendieck construction.
We further show that both of these model structures have the expected interactions with their $\infty$-counterparts; namely, with the contravariant model structure on $\mathrm{sSet}_{/ N\mathcal{C}}$ and with Lurie's cartesian model structure on $\mathrm{sSet}^+_{/ N\mathcal{C}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_11076 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A model structure for Grothendieck fibrations Moser, Lyne Sarazola, Maru Category Theory Algebraic Topology 18D30, 18N40, 18N60, 18A25 We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category $\mathcal{C}$. For the discrete case, we build a model structure on the slice $\mathrm{Cat}_{/\mathcal{C}}$, Quillen equivalent to the projective model structure on $[\mathcal{C}^{\mathrm{op}},\mathrm{Set}]$ via the classical category of elements construction. The cartesian case requires the use of markings, and we define a model structure on the slice $\mathrm{Cat}^+_{/\mathcal{C}}$, Quillen equivalent to the projective model structure on $[\mathcal{C}^{\mathrm{op}},\mathrm{Cat}]$ via a marked version of the Grothendieck construction. We further show that both of these model structures have the expected interactions with their $\infty$-counterparts; namely, with the contravariant model structure on $\mathrm{sSet}_{/ N\mathcal{C}}$ and with Lurie's cartesian model structure on $\mathrm{sSet}^+_{/ N\mathcal{C}}$. |
| title | A model structure for Grothendieck fibrations |
| topic | Category Theory Algebraic Topology 18D30, 18N40, 18N60, 18A25 |
| url | https://arxiv.org/abs/2306.11076 |