The Diffeological Čech-de Rham Obstruction
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914643636649984 |
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| author | Minichiello, Emilio |
| author_facet | Minichiello, Emilio |
| contents | Using higher topos theory, we explore the obstruction to the Čech-de Rham map being an isomorphism in each degree for diffeological spaces. In degree 1, we obtain an exact sequence which interprets Iglesias-Zemmour's construction from "Čech-de Rham Bicomplex in Diffeology" in $\infty$-stack cohomology. We obtain new exact sequences in all higher degrees. These exact sequences are constructed using homotopy pullback diagrams that include the $\infty$-stack classifying higher $\mathbb{R}$-bundle gerbes with connection. We also obtain a conceptual and succinct proof that the $\infty$-stack cohomology of the irrational torus $T_K$ for $K \subset \mathbb{R}$ a diffeologically discrete subgroup, agrees with the group cohomology of $K$ with values in $\mathbb{R}$. Finally, for a Lie group $G$, we prove that the groupoid of diffeological principal $G$-bundles with connection one obtains via higher topos theory is equivalent to the groupoid of diffeological principal $G$-bundles with connection defined in Waldorf's "Transgression to Loop Spaces and its Inverse, I". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Diffeological Čech-de Rham Obstruction Minichiello, Emilio Differential Geometry Algebraic Topology Category Theory Using higher topos theory, we explore the obstruction to the Čech-de Rham map being an isomorphism in each degree for diffeological spaces. In degree 1, we obtain an exact sequence which interprets Iglesias-Zemmour's construction from "Čech-de Rham Bicomplex in Diffeology" in $\infty$-stack cohomology. We obtain new exact sequences in all higher degrees. These exact sequences are constructed using homotopy pullback diagrams that include the $\infty$-stack classifying higher $\mathbb{R}$-bundle gerbes with connection. We also obtain a conceptual and succinct proof that the $\infty$-stack cohomology of the irrational torus $T_K$ for $K \subset \mathbb{R}$ a diffeologically discrete subgroup, agrees with the group cohomology of $K$ with values in $\mathbb{R}$. Finally, for a Lie group $G$, we prove that the groupoid of diffeological principal $G$-bundles with connection one obtains via higher topos theory is equivalent to the groupoid of diffeological principal $G$-bundles with connection defined in Waldorf's "Transgression to Loop Spaces and its Inverse, I". |
| title | The Diffeological Čech-de Rham Obstruction |
| topic | Differential Geometry Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2401.09400 |