A Quillen model structure of local homotopy equivalences
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866911817149710336 |
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| author | Mukherjee, Devarshi Cortiñas, Guillermo |
| author_facet | Mukherjee, Devarshi Cortiñas, Guillermo |
| contents | In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring $V$, the homotopy category of this model structure is the derived category of the quasi-abelian category $\overleftarrow{\mathsf{Ind}(\mathsf{Ban}_F)}$. This homotopy category is the appropriate target of the local and analytic cyclic homology theories for complete, torsionfree $V$-algebras and $\mathbb{F}$-algebras. When the base field is $\mathbb{C}$, the homotopy category is the target of local and analytic cyclic homology for pro-bornological $\mathbb{C}$-algebras, which includes the subcategory of pro-$C^*$-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_02979 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Quillen model structure of local homotopy equivalences Mukherjee, Devarshi Cortiñas, Guillermo K-Theory and Homology Algebraic Topology Category Theory In this note, we construct a closed model structure on the category of $\mathbb{Z}/2\mathbb{Z}$-graded complexes of projective systems of ind-Banach spaces. When the base field is the fraction field $F$ of a complete discrete valuation ring $V$, the homotopy category of this model structure is the derived category of the quasi-abelian category $\overleftarrow{\mathsf{Ind}(\mathsf{Ban}_F)}$. This homotopy category is the appropriate target of the local and analytic cyclic homology theories for complete, torsionfree $V$-algebras and $\mathbb{F}$-algebras. When the base field is $\mathbb{C}$, the homotopy category is the target of local and analytic cyclic homology for pro-bornological $\mathbb{C}$-algebras, which includes the subcategory of pro-$C^*$-algebras. |
| title | A Quillen model structure of local homotopy equivalences |
| topic | K-Theory and Homology Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2207.02979 |