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

(∞,1)-category theory

Background

• category theory

• higher category theory

• (n,r)-category

Basic concepts

• (∞,1)-category

• hom-objects

• equivalences in/of $(\infty,1)$-categories

• sub-(∞,1)-category

  • reflective sub-(∞,1)-category

  • reflective localization

• opposite (∞,1)-category

• over (∞,1)-category

  • join of quasi-categories

• (∞,1)-functor

  • exact (∞,1)-functor

  • (∞,1)-category of (∞,1)-functors

  • (∞,1)-category of (∞,1)-presheaves

• fibrations

  • inner fibration

  • left/right fibration

  • Cartesian fibration

    • Cartesian morphism

Universal constructions

• limit

  • terminal object

• adjoint functors

Local presentation

• locally presentable

  • essentially small

  • locally small

  • accessible

  • idempotent-complete

Theorems

• (∞,1)-Yoneda lemma

• (∞,1)-Grothendieck construction

• adjoint (∞,1)-functor theorem

• (∞,1)-monadicity theorem

Extra stuff, structure, properties

• stable (∞,1)-category

• (∞,1)-topos

Models

• category with weak equivalences

• model category

• derivator

• quasi-category

  • model structure for quasi-categories

  • model structure for Cartesian fibrations

• relation to simplicial categories

  • homotopy coherent nerve

  • simplicial model category

  • presentable quasi-category

• Kan complex

  • model structure for Kan complexes