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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2408.11157 |
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| _version_ | 1866916844868206592 |
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| author | Getzler, Ezra |
| author_facet | Getzler, Ezra |
| contents | We construct a natural morphism $ρ$ from the nerve $\text{MC}_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L)$ of a pronilpotent curved L${}_\infty$-algebra $L$ to the simplicial subset $γ_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L,s_\bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $γ_\bullet(L) \hookrightarrow \text{MC}_\bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_\infty$-algebras to curved L${}_\infty$-algebras. The morphism $ρ$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $ρ^\square$ of $ρ$ to identify $ρ$ with higher holonomy for semiabelian curved \Linf-algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11157 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher holonomy for curved L${}_\infty$-algebras 1: simplicial methods Getzler, Ezra Algebraic Topology Category Theory 18N40 (Primary) 18N45, 17B99 (Secondary) We construct a natural morphism $ρ$ from the nerve $\text{MC}_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L)$ of a pronilpotent curved L${}_\infty$-algebra $L$ to the simplicial subset $γ_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L,s_\bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $γ_\bullet(L) \hookrightarrow \text{MC}_\bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_\infty$-algebras to curved L${}_\infty$-algebras. The morphism $ρ$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $ρ^\square$ of $ρ$ to identify $ρ$ with higher holonomy for semiabelian curved \Linf-algebras. |
| title | Higher holonomy for curved L${}_\infty$-algebras 1: simplicial methods |
| topic | Algebraic Topology Category Theory 18N40 (Primary) 18N45, 17B99 (Secondary) |
| url | https://arxiv.org/abs/2408.11157 |