Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity
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arXiv
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| Autori principali: | , , , , , , , , |
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| Natura: | Preprint |
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2020
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| author | Calmès, Baptiste Dotto, Emanuele Harpaz, Yonatan Hebestreit, Fabian Land, Markus Moi, Kristian Nardin, Denis Nikolaus, Thomas Steimle, Wolfgang |
| author_facet | Calmès, Baptiste Dotto, Emanuele Harpaz, Yonatan Hebestreit, Fabian Land, Markus Moi, Kristian Nardin, Denis Nikolaus, Thomas Steimle, Wolfgang |
| contents | We define Grothendieck-Witt spectra in the setting of Poincaré $\infty$-categories and show that they fit into an extension with a K- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible. Our set-up allows for the uniform treatment of such algebraic examples alongside homotopy-theoretic generalisations: For example, the periodicity theorem holds for complex oriented $\mathrm{E}_1$-rings, and we show that the Grothendieck-Witt theory of parametrised spectra recovers Weiss and Williams' LA-theory.
Our Grothendieck-Witt spectra are defined via a version of the hermitian Q-construction, and a novel feature of our approach is to interpret the latter as a cobordism category. This perspective also allows us to give a hermitian version -- along with a concise proof -- of the theorem of Blumberg, Gepner and Tabuada, and provides a cobordism theoretic description of the aforementioned LA-spectra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2009_07224 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity Calmès, Baptiste Dotto, Emanuele Harpaz, Yonatan Hebestreit, Fabian Land, Markus Moi, Kristian Nardin, Denis Nikolaus, Thomas Steimle, Wolfgang K-Theory and Homology Algebraic Topology Category Theory 11E70, 18F25 (Primary), 11E39, 11E81, 19D25 (Secondary) We define Grothendieck-Witt spectra in the setting of Poincaré $\infty$-categories and show that they fit into an extension with a K- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible. Our set-up allows for the uniform treatment of such algebraic examples alongside homotopy-theoretic generalisations: For example, the periodicity theorem holds for complex oriented $\mathrm{E}_1$-rings, and we show that the Grothendieck-Witt theory of parametrised spectra recovers Weiss and Williams' LA-theory. Our Grothendieck-Witt spectra are defined via a version of the hermitian Q-construction, and a novel feature of our approach is to interpret the latter as a cobordism category. This perspective also allows us to give a hermitian version -- along with a concise proof -- of the theorem of Blumberg, Gepner and Tabuada, and provides a cobordism theoretic description of the aforementioned LA-spectra. |
| title | Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity |
| topic | K-Theory and Homology Algebraic Topology Category Theory 11E70, 18F25 (Primary), 11E39, 11E81, 19D25 (Secondary) |
| url | https://arxiv.org/abs/2009.07224 |