Thomason cohomology and Quillen's Theorem A
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914436282843136 |
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| author | Kirtisoglu, Mehmet Yalcin, Ergun |
| author_facet | Kirtisoglu, Mehmet Yalcin, Ergun |
| contents | Given a functor $φ: \mathcal{C} \to \mathcal{D}$ between two small categories, there is a homotopy equivalence $κ: hocolim _{\mathcal{D}} N(φ/-) \to N\mathcal{C}$ where $N(φ/-)$ is the functor which sends every object $d$ in $\mathcal{D}$ to the nerve of the comma category $φ/d$. We prove that the homotopy equivalence $κ$ induces an isomorphism on cohomology with coefficients in any coefficient system. As a consequence, we obtain a version of Quillen's Theorem A for the Thomason cohomology of categories. We also construct a spectral sequence for the Thomason cohomology of the Grothendieck construction $\int _{\mathcal{D}} F$ of a functor $F: \mathcal{D} \to Cat$ using the isomorphism in the main theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_14659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Thomason cohomology and Quillen's Theorem A Kirtisoglu, Mehmet Yalcin, Ergun Algebraic Topology Category Theory Primary: 18G90, Secondary: 55U10, 18G30, 18G35, 18G40 Given a functor $φ: \mathcal{C} \to \mathcal{D}$ between two small categories, there is a homotopy equivalence $κ: hocolim _{\mathcal{D}} N(φ/-) \to N\mathcal{C}$ where $N(φ/-)$ is the functor which sends every object $d$ in $\mathcal{D}$ to the nerve of the comma category $φ/d$. We prove that the homotopy equivalence $κ$ induces an isomorphism on cohomology with coefficients in any coefficient system. As a consequence, we obtain a version of Quillen's Theorem A for the Thomason cohomology of categories. We also construct a spectral sequence for the Thomason cohomology of the Grothendieck construction $\int _{\mathcal{D}} F$ of a functor $F: \mathcal{D} \to Cat$ using the isomorphism in the main theorem. |
| title | Thomason cohomology and Quillen's Theorem A |
| topic | Algebraic Topology Category Theory Primary: 18G90, Secondary: 55U10, 18G30, 18G35, 18G40 |
| url | https://arxiv.org/abs/2503.14659 |