nLab
derived ∞-Lie algebroid

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A derived \infty-Lie algebroid is an ∞-Lie algebroid whose underlying space is a derived space , for instance a derived smooth manifold.

More precisely, let TT be any abelian Lawvere theory and TCTAlg opT \hookrightarrow C \hookrightarrow T Alg^{op} a small full subcategory of geometric test objects formally dual to T-algebras. Then by the theory of function algebras on ∞-stacks over CC one identifies LieAlgd\infty LieAlgd with the subcategory

(TAlg Δ) op𝒪[C op,sSet] (T Alg^\Delta)^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} [C^{op}, sSet]

of the (∞,1)-category of (∞,1)-sheaves on CC modeled by cosimplicial T-algebras. Under the Dold-Kan correspondence applied to the underlying cosimplicial ordinary algbra, this identifies with non-negatively graded dg-algebras: the Chevalley-Eilenberg algebras of the corresponding dual \infty-Lie algebroids.

But in full derived geometry this setup is further generalized: instead of considering ∞-stacks on just a category of duals of TT-algebras, one considers derived ∞-stacks over an (∞,1)-site of simplicial algebras. Let TC Δ opTAlg Δ opT \hookrightarrow C^{\Delta^{op}} \hookrightarrow T Alg^{\Delta^{op}} be accordingly a small subcategory of simplicial TT-algebra, then the above adjunction generalizes to

((TAlg Δ op) Δ) op𝒪[C op,sSet], ((T Alg^{\Delta^{op}})^\Delta)^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\to} [C^{op}, sSet] \,,

where now on the left we have cosimplicial simplicial algebras . Under the Dold-Kan correspondence these now identify with unbounded dg-algebras CE(𝔞)CE(\mathfrak{a}). We have that

  • the categorical degree kk coming from k-morphisms of the \infty-sheaves contributes to positive degrees in CE(𝔞)CE(\mathfrak{a});

  • the derived degree ll coming from ll-morphisms in the \infty-function algebras ontribute to negative degree.

This category (TAlg Δ op×Δ) op(T Alg^{\Delta^{op} \times \Delta})^{op} we identify with that of derived \infty-Lie algebroids.

Examples

BV-BRST complex

In the literature the most familiar example of a derived \infty-Lie algebroids – even if not under this name – are the BV-BRST complexes. These are action Lie algebroids for actions of ∞-Lie algebras on derived smooth manifolds.

Revised on September 20, 2011 16:50:15 by Urs Schreiber (82.113.99.55)