Natural Transformations

Maps between functors: one arrow per object, all of them fitting into commuting squares — the precise meaning of a construction with no arbitrary choices.

The idea

A natural transformation is a map between two functors $F, G \colon \mathcal{C} \to \mathcal{D}$ with the same source and the same target, and it makes precise what it means for a construction to involve no arbitrary choices.

To illustrate what the definition is meant to capture, consider the determinant. It can be computed over the integers or over the integers modulo $n$, and reducing the entries and then taking the determinant gives the same answer as taking the determinant and then reducing — because the determinant is one fixed polynomial in the entries, used at every ring, with no ring receiving a formula of its own. Calling a construction natural means exactly this: one uniform recipe, with no arbitrary choice made anywhere.

Definition (Natural transformation).

Let $F, G \colon \mathcal{C} \to \mathcal{D}$ be functors. A natural transformation $\alpha \colon F \Rightarrow G$ is a family of arrows of $\mathcal{D}$ $\alpha_{A} \colon F(A) \to G(A),$ one for each object $A$ of $\mathcal{C}$ — its components — such that for every arrow $f \colon A \to A'$ of $\mathcal{C}$, $\alpha_{A'} \circ F(f) = G(f) \circ \alpha_{A}.$

The two sides are the two routes from $F(A)$ to $G(A')$, and the equation is called the naturality square. Nothing else is required.

This is the definition the subject was built to state: Eilenberg and Mac Lane invented categories and functors in order to say what natural means.

Ways to work on it

• Walkthrough. Learn what makes a family of maps between two functors natural, through examples and a non-example.
• Proof. Why a vector space is naturally isomorphic to its double dual but not to its single dual.
• Practice. Test families of maps for naturality and count the natural transformations in small cases.
• Hardest. Count the natural transformations between familiar functors and decide when pointwise isomorphisms are natural.

Builds on

• Functors

Leads to

• Adjoint Functors
• Equivalence of Categories
• Representable Functors

Not sure where to start? Take the ten-question placement test.