Homotopical Algebra for Lie Algebroids

Joost Nuiten

doi:10.1007/s10485-019-09563-z

Homotopical Algebra for Lie Algebroids

Joost Nuiten^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and L_∞-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.

Original language	English
Pages (from-to)	493-534
Number of pages	42
Journal	Applied Categorical Structures
Volume	27
Issue number	5
DOIs	https://doi.org/10.1007/s10485-019-09563-z
Publication status	Published - 1 Oct 2019

Funding

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).

Keywords

Dg-Lie algebroid
Lie algebroid cohomology
Model category

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Nuiten2019_Article_HomotopicalAlgebraForLieAlgebrFinal published version, 581 KBLicence: Taverne

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AB - We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and L∞-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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Homotopical Algebra for Lie Algebroids

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