André-Quillen Cohomology and $k$-invariants of simplicial categories
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916670157619200 |
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| author | Blanc, David Meadows, Nicholas |
| author_facet | Blanc, David Meadows, Nicholas |
| contents | Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the André-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$ take value. Given a map of simplicial categories $ϕ:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of $\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to study the obstruction to lifting $ϕ$ to $P^{(n)}\mathcal{X}$. In particular, an explicit description of this obstruction for the boundary of a cube can be used to recover various higher homotopy invariants of $\mathcal{X}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01885 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | André-Quillen Cohomology and $k$-invariants of simplicial categories Blanc, David Meadows, Nicholas Algebraic Topology Category Theory 55S35 (primary), 55S45, 18M60 Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the André-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$ take value. Given a map of simplicial categories $ϕ:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of $\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to study the obstruction to lifting $ϕ$ to $P^{(n)}\mathcal{X}$. In particular, an explicit description of this obstruction for the boundary of a cube can be used to recover various higher homotopy invariants of $\mathcal{X}$. |
| title | André-Quillen Cohomology and $k$-invariants of simplicial categories |
| topic | Algebraic Topology Category Theory 55S35 (primary), 55S45, 18M60 |
| url | https://arxiv.org/abs/2212.01885 |