nLab differential graded manifold

Idea

The notion of differential graded manifold is a generalization of the notion of smooth manifold from ordinary geometry to higher geometry, specifically to dg-geometry. Typically it is taken to be the formal dual to a dgc-algebra which in degree-0 is the algebra of smooth functions on an ordinary smooth manifold.

Hence this is a graded manifold whose algebra of functions is equipped with a compatible differential.

Sometimes this is called an “NQ-supermanifold”.

Examples

• An L-∞ algebroid over a smooth manifold may be thought of as a dg-manifold concentrated in non-negative degree.

• A derived L-∞ algebroid may be thought of as a dg-manifold in arbitrary degree.

Related concepts

• derived smooth manifold

• dg-scheme