nLab
differential graded manifold

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

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.

Revised on October 5, 2017 06:07:42 by Urs Schreiber (77.56.177.247)