027 MOC Category Theory

up:: 020 MOC Mathematics

Category Theory has been half-jokingly called “abstract nonsense” for quite a while now. It can be seen less harshly when talking about its role in abstraction, which is undoubtedly what Mathematics is all about, in the grand scheme of things. A great video that elucidates this is The Mathematician’s Weapon | Category Theory and Why We Care 1.0, from channel Eyesomorphic.

Basic concepts

• Category
  • Opposite Category
• Hom-Set
• Yoneda Lemma
• Universal Property

Arrows

• Monomorphism
  • An injective function is a monomorphism in the category Set
• Epimorphism
  • A surjective function is an epimorphism in the category Set
• Isomorphism (Category)
• Functor
  • Contravariant Functor
  • Group Actions can be seen as functors
  • Groups can be seen as automorphisms in single-element categories
• Natural Transformation
• Presheaf

Universal objects

• Limit (Category) // Colimit
  • Product (Category) // Coproduct
  • Pullback // Pushout
  • Equalizer
• Initial Object
• Terminal Object
  • A terminal object is isomorphic to an initial object if there is a morphism from T to I
• Zero Object

Category Examples

• Single-Element Category
• Groupoid
• Set Category
• Vect Category
• Sheaf
• Topos