nLab graded manifold

Idea

Under the embedding of smooth manifolds into formal duals of R-algebras a smooth manifold $X$ is identified with the formal dual of its $\mathbb{R}$-algebra of smooth functions. Given then a $\mathbb{Z}$-graded module $\mathfrak{a}$ over $C^\infty(X)$ concentrated away from zero, the graded symmetric graded algebra

$$Sym_{C^\infty(X)}\left( \mathfrak{a} \right)$$

may naturally be thought of as the formal dual of a $\mathbb{Z}$-graded manifold.

If moreover $Sym_{C^\infty(X)}\left(\mathfrak{a}\right)$ is equipped with a differential that makes it a differential graded-commutative algebra, then the formal dual is called a differential graded manifold.

Related concepts

• higher differential geometry, derived geometry

• Lie infinity-algebroid

• BRST-complex, BV-complex