higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
∞-Lie theory (higher geometry)
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”.
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.