Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917047029465088 |
|---|---|
| author | Verstraete, Marvin |
| author_facet | Verstraete, Marvin |
| contents | The purpose of this memoir is to study pre-Lie algebras up to homotopy with divided powers, and to use this algebraic structure for the study of mapping spaces in the category of operads. We define a new notion of algebra called $ΓΛ\mathcal{PL}_\infty$-algebra which characterizes the notion of $Γ(\mathcal{P}re\mathcal{L}ie_\infty,-)$-algebra. We also define a notion of a Maurer-Cartan element in complete $ΓΛ\mathcal{PL}_\infty$-algebras which generalizes the classical definition in Lie algebras. We prove that for every complete brace algebra $A$, and for every $n\geq 0$, the tensor product $A\otimesΣN^*(Δ^n)$ is endowed with the structure of a complete $ΓΛ\mathcal{PL}_\infty$-algebra, and define the simplicial Maurer-Cartan set $\mathcal{MC}_\bullet(A)$ associated to $A$ as the Maurer-Cartan set of $ A\otimesΣN^*(Δ^\bullet)$. We compute the homotopy groups of this simplicial set, and prove that the functor $\mathcal{MC}_\bullet(-)$ satisfies a homotopy invariance result, which extends the Goldman-Millson theorem in dimension $0$. As an application, we give a description of mapping spaces in the category of non-symmetric operads in terms of this simplicial Maurer-Cartan set. We etablish a generalization of the latter result for symmetric operads. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces Verstraete, Marvin Algebraic Topology 18N50 18M70 17B55 55U35 The purpose of this memoir is to study pre-Lie algebras up to homotopy with divided powers, and to use this algebraic structure for the study of mapping spaces in the category of operads. We define a new notion of algebra called $ΓΛ\mathcal{PL}_\infty$-algebra which characterizes the notion of $Γ(\mathcal{P}re\mathcal{L}ie_\infty,-)$-algebra. We also define a notion of a Maurer-Cartan element in complete $ΓΛ\mathcal{PL}_\infty$-algebras which generalizes the classical definition in Lie algebras. We prove that for every complete brace algebra $A$, and for every $n\geq 0$, the tensor product $A\otimesΣN^*(Δ^n)$ is endowed with the structure of a complete $ΓΛ\mathcal{PL}_\infty$-algebra, and define the simplicial Maurer-Cartan set $\mathcal{MC}_\bullet(A)$ associated to $A$ as the Maurer-Cartan set of $ A\otimesΣN^*(Δ^\bullet)$. We compute the homotopy groups of this simplicial set, and prove that the functor $\mathcal{MC}_\bullet(-)$ satisfies a homotopy invariance result, which extends the Goldman-Millson theorem in dimension $0$. As an application, we give a description of mapping spaces in the category of non-symmetric operads in terms of this simplicial Maurer-Cartan set. We etablish a generalization of the latter result for symmetric operads. |
| title | Pre-Lie algebras up to homotopy with divided powers and homotopy of operadic mapping spaces |
| topic | Algebraic Topology 18N50 18M70 17B55 55U35 |
| url | https://arxiv.org/abs/2507.03632 |