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graded manifold

Contents

Context

Higher geometry

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

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

Sym C (X)(𝔞) 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 (X)(𝔞)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 06:06:38. See the history of this page for a list of all contributions to it.