Spectral sequences via linear presheaves
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866914414954807296 |
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| author | Livernet, Muriel Whitehouse, Sarah |
| author_facet | Livernet, Muriel Whitehouse, Sarah |
| contents | We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two décalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term décalage in connection with spectral sequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02777 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral sequences via linear presheaves Livernet, Muriel Whitehouse, Sarah Algebraic Topology Category Theory 18G40, 18N40 We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two décalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term décalage in connection with spectral sequences. |
| title | Spectral sequences via linear presheaves |
| topic | Algebraic Topology Category Theory 18G40, 18N40 |
| url | https://arxiv.org/abs/2406.02777 |