Morphisms between functors and the naturality square
Functors can themselves be connected by arrows. These natural transformations are the reason category theory was invented.
Press Animate and watch two paths travel around the square. They arrive at the same place, and that is naturality.
If functors are maps between categories, natural transformations are maps between functors. Saunders Mac Lane, co-inventor of category theory, said: "I didn't invent categories to study functors; I invented them to study natural transformations."
A natural transformation α : F ⇒ G assigns to each object A a morphism α_A : F(A) → G(A) such that the naturality square commutes. Watch both paths arrive at the same place!
Key insight: The naturality condition says: α_B ∘ F(f) = G(f) ∘ α_A for every morphism f. Both paths through the square give the same result — that's what makes the transformation "natural" (not dependent on arbitrary choices).
Pick a pair of matrices and check that det(AB) = det(A) · det(B) holds every time.
The determinant det : GL_n ⇒ (−)* is a natural transformation between functors on commutative rings! Each component is a group homomorphism — that is the identity det(AB) = det(A) · det(B) — and naturality says det commutes with every ring homomorphism φ: det(φ(A)) = φ(det(A)). Verify the homomorphism identity with concrete matrices below.
Key insight: The determinant is "natural" because it works uniformly over every coefficient ring: applying a ring homomorphism and then taking det gives the same answer as taking det first. This is a concrete example of the naturality square commuting: the two paths always give the same answer.
Click each natural transformation to see how it connects the functors, and how two of them compose.
Functors themselves form a category! In the functor category [C, D], objects are functors F, G, H : C → D and morphisms are natural transformations between them.
Key insight: Natural transformations compose component-wise: (β∘α)_A = β_A ∘ α_A. This gives us a whole new category whose "objects" are functors! This is one of the most powerful ideas in category theory.
Click through the examples. Each one works the same way everywhere, with no arbitrary choices.
The canonical embedding of a vector space into its double dual. Natural because it doesn't require choosing a basis.
Natural transformations appear throughout mathematics. Here are some famous examples that show why "naturality" is such a powerful concept.
Key insight: The common thread: all these transformations work "uniformly" — they don't depend on arbitrary choices. Eilenberg and Mac Lane invented category theory precisely to make the word "natural" mathematically precise!
Famous Examples — Double dual, Hurewicz, unit/counit — all are natural.
Spot a mistake or have an idea?