2-category theory

Definitions

• 2-category

• strict 2-category

• bicategory

• enriched bicategory

Transfors between 2-categories

• 2-functor

  • pseudofunctor

  • lax functor

  • equivalence of 2-categories

• 2-natural transformation

  • lax natural transformation

  • icon

• modification

• Yoneda lemma for bicategories

Morphisms in 2-categories

• fully faithful morphism

• faithful morphism

• conservative morphism

• pseudomonic morphism

• discrete morphism

• eso morphism

Structures in 2-categories

• adjunction

• mate

• monad

• cartesian object

• fibration in a 2-category

• codiscrete cofibration

Limits in 2-categories

• 2-limit

• 2-pullback

• comma object

• inserter

• inverter

• equifier

Structures on 2-categories

• 2-monad

  • lax-idempotent 2-monad

  • pseudomonad

  • pseudoalgebra for a 2-monad

• monoidal 2-category

  • cartesian bicategory

• Gray tensor product

• proarrow equipment

higher category theory

• category theory
• homotopy theory

Basic concepts

• k-morphism, coherence
• looping and delooping
• looping and suspension

Basic theorems

• homotopy hypothesis-theorem
• delooping hypothesis-theorem
• periodic table
• stabilization hypothesis-theorem
• exactness hypothesis
• holographic principle

Applications

• applications of (higher) category theory
• higher category theory and physics

Models

• (n,r)-category
  • Theta-space
  • ∞-category/∞-category
  • (∞,n)-category
    • n-fold complete Segal space
  • (∞,2)-category
  • (∞,1)-category
    • quasi-category
      • algebraic quasi-category
    • simplicially enriched category
    • complete Segal space
    • model category
  • (∞,0)-category/∞-groupoid
    • Kan complex
      • algebraic Kan complex
      • simplicial T-complex
  • n-category = (n,n)-category
    • 2-category, (2,1)-category
    • 1-category
    • 0-category
    • (-1)-category
    • (-2)-category
  • n-poset = (n-1,n)-category
    • poset = (0,1)-category
    • 2-poset = (1,2)-category
  • n-groupoid = (n,0)-category
    • 2-groupoid, 3-groupoid
• categorification/decategorification
• geometric definition of higher category
  • Kan complex
  • quasi-category
  • simplicial model for weak ∞-categories
    • complicial set
    • weak complicial set
• algebraic definition of higher category
  • bicategory
  • bigroupoid
  • tricategory
  • tetracategory
  • strict ∞-category
  • Batanin ∞-category
  • Trimble ∞-category
  • Grothendieck-Maltsiniotis ∞-categories
• stable homotopy theory
  • symmetric monoidal category
  • symmetric monoidal (∞,1)-category
  • stable (∞,1)-category
    • dg-category
    • A-∞ category
    • triangulated category

Morphisms

• k-morphism
  • 2-morphism
• transfor
  • natural transformation
  • modification

Functors

• functor
• 2-functor
  • pseudofunctor
  • lax functor
• (∞,1)-functor

Universal constructions

• 2-limit
• (∞,1)-adjunction
• (∞,1)-Kan extension
  • (∞,1)-limit
• (∞,1)-Grothendieck construction

Extra properties and structure

• cosmic cube
  • k-tuply monoidal n-category
  • strict ∞-category, strict ∞-groupoid
• stable (∞,1)-category
• (∞,1)-topos

1-categorical presentations

• homotopical category
• model category theory
• enriched category theory