Contents

### Context

Ingredients

Concepts

Constructions

Examples

Theorems

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## 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.

Created on October 5, 2017 at 10:06:38. See the history of this page for a list of all contributions to it.