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| Main Author: | |
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| Format: | Preprint |
| Published: |
2017
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1712.00521 |
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| _version_ | 1866929485064962048 |
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| author | Heine, Hadrian |
| author_facet | Heine, Hadrian |
| contents | For any motivic $\mathbb{E}_\infty$-ring spectrum $A$ we construct an equivalence $ρ$ between the $\infty$-category of cellular motivic $A$-module spectra and modules over an $\mathbb{E}_1$-algebra $Θ$ in $\mathbb{Z} $-graded spectra, under which the motivic grading corresponds to the $\mathbb{Z}$-grading. If the base is the complex numbers or if $A$ admits an $\mathbb{E}_\infty$-orientation, we refine the $\mathbb{E}_1$-algebra $Θ$ to an $\mathbb{E}_\infty$-algebra and $ρ$ to a symmetric monoidal equivalence. To capture the symmetric monoidal structure in the general situation, we lift $ρ$ to a symmetric monoidal equivalence to modules over an $\mathbb{E}_\infty$-algebra in $\mathcal{J} $-graded spectra that invert morphisms of $\mathcal{J}$, where $\mathcal{J}$ is the diagram category of Sagave-Schlichtkrull, a model for Quillen's localization of the groupoid of finite sets and bijections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1712_00521 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A topological model for cellular motivic spectra Heine, Hadrian Algebraic Topology For any motivic $\mathbb{E}_\infty$-ring spectrum $A$ we construct an equivalence $ρ$ between the $\infty$-category of cellular motivic $A$-module spectra and modules over an $\mathbb{E}_1$-algebra $Θ$ in $\mathbb{Z} $-graded spectra, under which the motivic grading corresponds to the $\mathbb{Z}$-grading. If the base is the complex numbers or if $A$ admits an $\mathbb{E}_\infty$-orientation, we refine the $\mathbb{E}_1$-algebra $Θ$ to an $\mathbb{E}_\infty$-algebra and $ρ$ to a symmetric monoidal equivalence. To capture the symmetric monoidal structure in the general situation, we lift $ρ$ to a symmetric monoidal equivalence to modules over an $\mathbb{E}_\infty$-algebra in $\mathcal{J} $-graded spectra that invert morphisms of $\mathcal{J}$, where $\mathcal{J}$ is the diagram category of Sagave-Schlichtkrull, a model for Quillen's localization of the groupoid of finite sets and bijections. |
| title | A topological model for cellular motivic spectra |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/1712.00521 |