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 algebra

universal algebra

Algebraic theories

• algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory

• monad / (∞,1)-monad

• operad / (∞,1)-operad

Algebras and modules

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

• action, ∞-action

• representation, ∞-representation

• module, ∞-module

• associated bundle, associated ∞-bundle

Higher algebras

• monoidal (∞,1)-category

  • symmetric monoidal (∞,1)-category

  • monoid in an (∞,1)-category

  • commutative monoid in an (∞,1)-category

• symmetric monoidal (∞,1)-category of spectra

  • smash product of spectra

  • symmetric monoidal smash product of spectra

  • ring spectrum, module spectrum, algebra spectrum

• A-∞ algebra

  • A-∞ ring, A-∞ space

• C-∞ algebra

• E-∞ ring, E-∞ algebra

  • ∞-module, (∞,1)-module bundle

  • multiplicative cohomology theory

• L-∞ algebra

  • deformation theory

Model category presentations

• model structure on simplicial T-algebras / homotopy T-algebra

• model structure on operads

  model structure on algebras over an operad

Geometry on formal duals of algebras

• Isbell duality

• derived geometry

Theorems

• Deligne conjecture

• delooping hypothesis

• monoidal Dold-Kan correspondence