On the cohomology of differentiable stacks
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910631203962880 |
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| author | del Hoyo, Matias Ortiz, Cristian Studzinski, Fernando |
| author_facet | del Hoyo, Matias Ortiz, Cristian Studzinski, Fernando |
| contents | Morita equivalence classes of Lie groupoids serve as models for differentiable stacks, which are higher spaces in differential geometry, generalizing manifolds and orbifolds. Representations up to homotopy of Lie groupoids provide a higher analog of classical representations and play a significant role in Poisson geometry. In this paper, we prove that the cohomology with coefficients in a representation up to homotopy is a Morita invariant, and therefore an invariant of the underlying stack. This result was inspired by the 2-term case, previously developed by del Hoyo and Ortiz, and it relies on the simplicial approach to representations up to homotopy, recently introduced by del Hoyo and Trentinaglia. As a subsidiary result, we include a proof of the cohomological descent for higher Lie groupoids. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_02570 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the cohomology of differentiable stacks del Hoyo, Matias Ortiz, Cristian Studzinski, Fernando Differential Geometry Algebraic Topology Category Theory Representation Theory 22A22, 58H05, 18N50, 18G35 Morita equivalence classes of Lie groupoids serve as models for differentiable stacks, which are higher spaces in differential geometry, generalizing manifolds and orbifolds. Representations up to homotopy of Lie groupoids provide a higher analog of classical representations and play a significant role in Poisson geometry. In this paper, we prove that the cohomology with coefficients in a representation up to homotopy is a Morita invariant, and therefore an invariant of the underlying stack. This result was inspired by the 2-term case, previously developed by del Hoyo and Ortiz, and it relies on the simplicial approach to representations up to homotopy, recently introduced by del Hoyo and Trentinaglia. As a subsidiary result, we include a proof of the cohomological descent for higher Lie groupoids. |
| title | On the cohomology of differentiable stacks |
| topic | Differential Geometry Algebraic Topology Category Theory Representation Theory 22A22, 58H05, 18N50, 18G35 |
| url | https://arxiv.org/abs/2410.02570 |