Poincaré Duality Pairs of $\infty$-Categories
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912666419724288 |
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| author | Bianchi, Andrea Hilman, Kaif Kirstein, Dominik Kremer, Christian |
| author_facet | Bianchi, Andrea Hilman, Kaif Kirstein, Dominik Kremer, Christian |
| contents | We introduce a notion of Poincaré duality for pairs of $\infty$-categories, extending Poincaré-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a uniform treatment of Wall's Poincaré ads of spaces, iterated Poincaré cobordisms, and in general, diagrams of spaces parametrised by the face poset of a combinatorial manifold. In each of these cases, the theory reduces them to studying a single pair of $\infty$-categories and the properties of a single functor, the relative cohomology functor. Using this formalism, we prove a very general fibration theorem which, in particular, specialises to a generalisation of Klein-Qin-Su's fibration theorem for Poincaré triads to all ads. This theory also lays the foundation for future work by the authors on Poincaré cobordism categories, isovariant Poincaré spaces and string topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20646 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poincaré Duality Pairs of $\infty$-Categories Bianchi, Andrea Hilman, Kaif Kirstein, Dominik Kremer, Christian Algebraic Topology Category Theory Geometric Topology 16D90, 55N20, 57P10, 57R19 We introduce a notion of Poincaré duality for pairs of $\infty$-categories, extending Poincaré-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a uniform treatment of Wall's Poincaré ads of spaces, iterated Poincaré cobordisms, and in general, diagrams of spaces parametrised by the face poset of a combinatorial manifold. In each of these cases, the theory reduces them to studying a single pair of $\infty$-categories and the properties of a single functor, the relative cohomology functor. Using this formalism, we prove a very general fibration theorem which, in particular, specialises to a generalisation of Klein-Qin-Su's fibration theorem for Poincaré triads to all ads. This theory also lays the foundation for future work by the authors on Poincaré cobordism categories, isovariant Poincaré spaces and string topology. |
| title | Poincaré Duality Pairs of $\infty$-Categories |
| topic | Algebraic Topology Category Theory Geometric Topology 16D90, 55N20, 57P10, 57R19 |
| url | https://arxiv.org/abs/2510.20646 |