nLab
derived ∞-Lie algebroid

Idea

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

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

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

of the (∞,1)-category of (∞,1)-sheaves on $C$ 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 $\mathrm{\infty }$-Lie algebroids.

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

$$((T{\mathrm{Alg}}^{{\Delta }^{\mathrm{op}}}{)}^{\Delta }{)}^{\mathrm{op}}\stackrel{\stackrel{𝒪}{\leftarrow }}{\to }[C^{\mathrm{op}},\mathrm{sSet}],$$((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 $\mathrm{CE}(𝔞)$. We have that

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

• the derived degree $l$ coming from $l$-morphisms in the $\mathrm{\infty }$-function algebras ontribute to negative degree.

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

Examples

BV-BRST complex

In the literature the most familiar example of a derived $\mathrm{\infty }$-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.

Related concepts

• Lie algebra, Lie algebroid

• L-∞ algebra, L-∞ algebroid