More precisely, let be any abelian Lawvere theory and a small full subcategory of geometric test objects formally dual to T-algebras. Then by the theory of function algebras on ∞-stacks over one identifies with the subcategory
of the (∞,1)-category of (∞,1)-sheaves on 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 -Lie algebroids.
But in full derived geometry this setup is further generalized: instead of considering ∞-stacks on just a category of duals of -algebras, one considers derived ∞-stacks over an (∞,1)-site of simplicial algebras. Let be accordingly a small subcategory of simplicial -algebra, then the above adjunction generalizes to
where now on the left we have cosimplicial simplicial algebras . Under the Dold-Kan correspondence these now identify with unbounded dg-algebras . We have that
the categorical degree coming from k-morphisms of the -sheaves contributes to positive degrees in ;
the derived degree coming from -morphisms in the -function algebras ontribute to negative degree.
This category we identify with that of derived -Lie algebroids.
In the literature the most familiar example of a derived -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.