A Deligne conjecture for prestacks
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929758041800704 |
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| author | Campos, Ricardo Hermans, Lander |
| author_facet | Campos, Ricardo Hermans, Lander |
| contents | We prove an analog of the Deligne conjecture for prestacks. We show that given a prestack $\mathbb A$, its Gerstenhaber--Schack complex $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$ is naturally an $E_2$-algebra. This structure generalises both the known $\mathsf{L}_\infty$-algebra structure on $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$, as well as the Gerstenhaber algebra structure on its cohomology $\mathbf{H}_{\mathsf{GS}}(\mathbb A)$. The main ingredient is the proof of a conjecture of Hawkins \cite{hawkins}, stating that the dg operad $\mathsf{Quilt}$ has vanishing homology in positive degrees. As a corollary, $\mathsf{Quilt}$ is quasi-isomorphic to the operad $\mathsf{Brace}$ encoding brace algebras. In addition, we improve the $L_\infty$-structure on $\mathsf{Quilt}$ by showing that it originates from a $\mathsf{PreLie}_\infty$-structure lifting the $\mathsf{PreLie}$-structure on $\mathsf{Brace}$ in homology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Deligne conjecture for prestacks Campos, Ricardo Hermans, Lander Algebraic Topology Algebraic Geometry Category Theory K-Theory and Homology Quantum Algebra We prove an analog of the Deligne conjecture for prestacks. We show that given a prestack $\mathbb A$, its Gerstenhaber--Schack complex $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$ is naturally an $E_2$-algebra. This structure generalises both the known $\mathsf{L}_\infty$-algebra structure on $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$, as well as the Gerstenhaber algebra structure on its cohomology $\mathbf{H}_{\mathsf{GS}}(\mathbb A)$. The main ingredient is the proof of a conjecture of Hawkins \cite{hawkins}, stating that the dg operad $\mathsf{Quilt}$ has vanishing homology in positive degrees. As a corollary, $\mathsf{Quilt}$ is quasi-isomorphic to the operad $\mathsf{Brace}$ encoding brace algebras. In addition, we improve the $L_\infty$-structure on $\mathsf{Quilt}$ by showing that it originates from a $\mathsf{PreLie}_\infty$-structure lifting the $\mathsf{PreLie}$-structure on $\mathsf{Brace}$ in homology. |
| title | A Deligne conjecture for prestacks |
| topic | Algebraic Topology Algebraic Geometry Category Theory K-Theory and Homology Quantum Algebra |
| url | https://arxiv.org/abs/2406.16652 |