Homotopical algebra for Lie algebroids
Joost Nuiten

TL;DR
This paper develops a homotopical algebra framework for dg-Lie algebroids and $L_$-algebroids, enabling advanced algebraic techniques to study their properties and cohomology.
Contribution
It constructs Quillen equivalent semi-model structures on categories of dg-Lie and $L_$-algebroids, facilitating homotopical methods in their analysis.
Findings
Lie algebroids can be resolved by dg-Lie algebroids from dg-Lie algebras
Lie algebroid cohomology is represented in the homotopy category
Homotopical algebra tools are applicable to dg-Lie algebroids
Abstract
We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and -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. that have a null-homotopic anchor map. 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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