All known realizations of complete Lie algebras coincide
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866909617121918976 |
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| author | Félix, Yves Fuentes, Mario Murillo, Aniceto |
| author_facet | Félix, Yves Fuentes, Mario Murillo, Aniceto |
| contents | We prove that for any reduced differential graded Lie algebra L, the classical Quillen geometrical realization $\langle L\rangle_Q$ is homotopy equivalent to the realization $\langle L\rangle= Hom_{\bf cdgl}(\mathfrak{L}_\bullet, L)$ constructed via the cosimplicial free complete differential graded Lie algebra $\mathfrak{L}_\bullet$. As the latter is a deformation retract of the Deligne-Getzler-Hinich realization MC${}_\bullet(L)$ we deduce that, up to homotopy, there is only one realization functor for complete differential graded Lie algebras. Immediate consequences include an elementary proof of the Baues-Lemaire conjecture and the description of the Quillen realization as a representable functor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_10886 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | All known realizations of complete Lie algebras coincide Félix, Yves Fuentes, Mario Murillo, Aniceto Algebraic Topology We prove that for any reduced differential graded Lie algebra L, the classical Quillen geometrical realization $\langle L\rangle_Q$ is homotopy equivalent to the realization $\langle L\rangle= Hom_{\bf cdgl}(\mathfrak{L}_\bullet, L)$ constructed via the cosimplicial free complete differential graded Lie algebra $\mathfrak{L}_\bullet$. As the latter is a deformation retract of the Deligne-Getzler-Hinich realization MC${}_\bullet(L)$ we deduce that, up to homotopy, there is only one realization functor for complete differential graded Lie algebras. Immediate consequences include an elementary proof of the Baues-Lemaire conjecture and the description of the Quillen realization as a representable functor. |
| title | All known realizations of complete Lie algebras coincide |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2207.10886 |