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Homotopical algebra for Lie algebroids

Joost Nuiten

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.

Homotopical algebra for 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.

Paper Structure

This paper contains 20 sections, 29 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Conventions
  4. Acknowledgements
  5. Preliminaries
  6. DG-Lie-algebroids
  7. $L_\infty$-algebroids
  8. Semi-model categories
  9. Main results
  10. Filtrations on cell attachments
  11. Filtrations
  12. Coproducts with $L_\infty$-algebras
  13. Pushouts along free maps
  14. Proof of Theorem \ref{['la:thm:modelstructureondglralgebras']}
  15. Proof of Theorem \ref{['la:thm:monadicity']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let $A$ be a commutative dg-algebra over a field of characteristic zero. The category of dg-Lie algebroids carries a semi-model structure, in which a map is a weak equivalence (fibration) if it is a quasi-isomorphism (degreewise surjective).

Theorems & Definitions (110)

  • Theorem
  • Remark : Global approach
  • Theorem
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4: cf. lod12
  • Remark 2.6
  • Definition 2.7: see e.g. bon13, pym16
  • Example 2.9
  • ...and 100 more