Abstract
We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and \(L_\infty \)-algebroids over a commutative dg-algebra in characteristic zero. This allows one to apply the usual methods of homotopical algebra to dg-Lie algebroids: for example, every Lie algebroid can be resolved by dg-Lie algebroids that arise from dg-Lie algebras, i.e. whose anchor map is zero. As an application, we show how Lie algebroid cohomology is represented by an object in the homotopy category of dg-Lie algebroids.
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Acknowledgements
I would like to thank the anonymous referee, whose many questions and comments have helped greatly improving the clarity and exposition of the paper. This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO).
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Communicated by Vladimir Hinich.
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Nuiten, J. Homotopical Algebra for Lie Algebroids. Appl Categor Struct 27, 493–534 (2019). https://doi.org/10.1007/s10485-019-09563-z
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DOI: https://doi.org/10.1007/s10485-019-09563-z