∞-Lie theory (higher geometry)

Background

Smooth structure

• generalized smooth space

  • smooth manifold

  • diffeological space

  • Frölicher space

• smooth topos

  • Cahiers topos

• smooth ∞-groupoid, concrete smooth ∞-groupoid

• synthetic differential ∞-groupoid

Higher groupoids

• ∞-groupoid

  • groupoid

  • 2-groupoid

  • strict ∞-groupoid

    • crossed complex

• ∞-group

  • simplicial group

Lie theory

• Lie theory

  • Lie integration, Lie differentiation

  • Lie's three theorems

  • Lie theory for stacky Lie groupoids

∞-Lie groupoids

• ∞-Lie groupoid

  • strict ∞-Lie groupoid

  • Lie groupoid

    • differentiable stack

    • orbifold

• ∞-Lie group

  • Lie group

    • simple Lie group, semisimple Lie group

  • Lie 2-group

∞-Lie algebroids

• ∞-Lie algebroid

  • Lie algebroid

• Lie ∞-algebroid representation

• L-∞-algebra

  • model structure for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras

  • Lie algebra

    • semisimple Lie algebra, compact Lie algebra

  • Lie 2-algebra

    • strict Lie 2-algebra

    • differential crossed module

• Lie 3-algebra

  • differential 2-crossed module

• dg-Lie algebra, simplicial Lie algebra

• super L-∞ algebra

  • super Lie algebra

Formal Lie groupoids

• formal group, formal groupoid

Cohomology

• Lie algebra cohomology

  • Chevalley-Eilenberg algebra

  • Weil algebra

  • invariant polynomial

    • Killing form

• nonabelian Lie algebra cohomology

Homotopy

• homotopy groups of a Lie groupoid

Related topics

• ∞-Chern-Weil theory

Examples

$\infty$-Lie groupoids

• Atiyah Lie groupoid

• fundamental ∞-groupoid

  • path groupoid

  • path n-groupoid

• smooth principal ∞-bundle

$\infty$-Lie groups

• orthogonal group

  • special orthogonal group

  • spin group

  • string 2-group

  • fivebrane 6-group

• unitary group

  • special unitary group

• circle Lie n-group

  • circle group

$\infty$-Lie algebroids

• tangent Lie algebroid

• action Lie algebroid

• Atiyah Lie algebroid

• symplectic Lie n-algebroid

  • symplectic manifold

  • Poisson Lie algebroid

  • Courant Lie algebroid

    • generalized complex geometry

$\infty$-Lie algebras

• general linear Lie algebra

• orthogonal Lie algebra, special orthogonal Lie algebra

• endomorphism L-∞ algebra

• automorphism ∞-Lie algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra

• supergravity Lie 3-algebra

• supergravity Lie 6-algebra

• line Lie n-algebra