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# 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