Homotopical algebra for Lie algebroids

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 $L_\infty$-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. 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.