Category Theory Illustrated – Natural Transformations
6 hours ago
- #functors
- #natural-transformations
- #category-theory
- Natural transformations are central to category theory, introduced to study morphisms between functors.
- They enable the definition of category equality and advanced concepts like equivalent categories.
- Natural transformations are viewed as mappings between functors, involving object and morphism mappings.
- The naturality condition ensures that transformations commute with functor applications, forming commuting squares.
- In programming, natural transformations correspond to polymorphic functions, linking types and functors.
- Natural transformations compose both vertically and horizontally, forming a higher-level structure in category theory.
- The category of small categories (Cat) uses functors as morphisms and natural transformations as 2-morphisms.
- Equivalent categories are defined via natural isomorphisms between functors, relaxing strict equality to isomorphism.