Spin(7)/G₂ is the 7-sphere in nLab

group theory

• group, ∞-group
• group object, group object in an (∞,1)-category
• abelian group, spectrum
• super abelian group
• group action, ∞-action
• representation, ∞-representation
• progroup
• homogeneous space

Classical groups

• general linear group

• unitary group

  • special unitary group. projective unitary group

• orthogonal group

  • special orthogonal group

• symplectic group

Finite groups

• finite group

• symmetric group, cyclic group, braid group

• classification of finite simple groups

• sporadic finite simple groups

  • Monster group, Mathieu group

Group schemes

• algebraic group

• abelian variety

Topological groups

• topological group

• compact topological group, locally compact topological group

• maximal compact subgroup

• string group

Lie groups

• Lie group

• compact Lie group

• Kac-Moody group

Super-Lie groups

• super Lie group

• super Euclidean group

Higher groups

• 2-group

  • crossed module, strict 2-group

• n-group

• ∞-group

  • simplicial group

  • crossed complex

  • k-tuply groupal n-groupoid

  • spectrum

• circle n-group, string 2-group, fivebrane Lie 6-group

Cohomology and Extensions

• group cohomology

• group extension,

• ∞-group extension, Ext-group

Related concepts

• quantum group

higher geometry / derived geometry

Ingredients

• higher topos theory

• higher algebra

Concepts

• geometric little (∞,1)-toposes

  • structured (∞,1)-topos

  • geometry (for structured (∞,1)-toposes)

  • generalized scheme

• geometric big (∞,1)-toposes

  • cohesive (∞,1)-topos

• function algebras on ∞-stacks

  • geometric ∞-stacks

Constructions

• loop space object, free loop space object

• fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos

Examples

• derived algebraic geometry

  • étale (∞,1)-site, Hochschild cohomology of dg-algebras

  • dg-geometry

    • dg-scheme

  • schematic homotopy type

• derived noncommutative geometry

  • noncommutative geometry

• derived smooth geometry

  • differential geometry, differential topology

  • derived smooth manifold, dg-manifold

  • smooth ∞-groupoid, ∞-Lie algebroid

• higher symplectic geometry

• higher Klein geometry

• higher Cartan geometry

Theorems

• Isbell duality

• Jones' theorem, Deligne-Kontsevich conjecture

• Tannaka duality for geometric stacks