category theory

Concepts

• category

• functor

• natural transformation

• Cat

Universal constructions

• universal construction

  • representable functor

  • adjoint functor

  • limit/colimit

  • weighted limit

  • end/coend

  • Kan extension

Theorems

• Yoneda lemma

• Isbell duality

• Grothendieck construction

• adjoint functor theorem

• monadicity theorem

• adjoint lifting theorem

• Tannaka duality

• Gabriel-Ulmer duality

• small object argument

• Freyd-Mitchell embedding theorem

• relation between type theory and category theory

Extensions

• sheaf and topos theory

• enriched category theory

• higher category theory

Applications

• applications of (higher) category theory

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/ω-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 ω-categories

    • complicial set

    • weak complicial set

• algebraic definition of higher category

  • bicategory

  • bigroupoid

  • tricategory

  • tetracategory

  • strict ω-category

  • Batanin ω-category

  • Trimble ω-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