Category Theory: OUTLINE

Category theory

• Prerequisite

  • sets

  • relations

  • functions

    • identity

    • assoc

    • injective, surjective, bijective

    • inverse

  • abstract algebra

    • algebraic structures

      • algebraic structure

        • carrier set

        • binary operation over the set

        • that together uphold a set of axioms

    • group-like structures

      • magma

      • sg

      • monoid

      • group

      • Abelian group

  • Commutative diagram

• Basic concepts

  • category, 𝒞

  • objects, Obj(𝒞)

  • morphism/arrow, Arr(𝒞)

  • composition of arrows, (∘)

  • identity as unit of composition, (1ᴀ)

  • transitive closure

  • duality

    • opposite category

    • opposite functor

  • hom-set

  • natural transformation

• Categories

  • category

    • category element

    • objects

      • initial object

      • terminal object

  • opposite category

  • concrete category

  • CCC, cartesian closed category

  • closed monoidal category

  • Kleisli category

  • monoid

  • groupoid category in which all morphisms are invertible

  • monad

• Morphisms

  • arrow

  • isomorphism

  • automorphism

  • endomorphism

  • epimorphism, epic

  • monomorphism, monic

  • homomorphism

  • homeomorphism

• Functors

  • functor, covariant functor

  • contravariant functor

  • bifunctor

  • profunctor

  • multifunctor

  • endofunctor

  • applicative functor

  • opposite functor

  • adjunction - relation on functors

    • adjoint functor

  • Exponential functor

  • representable functor

    Anafunctors

• Essential concepts

  • category

  • functor

  • natural transformation

• Universal constructions

  • Natural transformation

  • Universal property

  • universal construction

  • representable functor

  • CCC

  • adjunction

    • adjoint functor

    • right adjoint

    • left adjoint

  • co/limits

    • product

    • coproduct

    • limit

    • colimit

    • weighted limit

  • co/end

    • end

    • coend

  • Kan extension

  • 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

• CT extensions

  • sheaf and topos theory

  • enriched category theory

  • higher category theory

The Yoneda lemma naturality naturality formula naturality condition groupoid symmetric groupoid Anafunctors