Natural transformations

Builds on

Required prerequisites, all the way down: 16

You will learn to

• Verify naturality for examples such as V → V** and the determinant GLₙ(R) → Rˣ, natural in the commutative ring R.
• Explain why V ≅ V** is natural, while V ≅ V* cannot even be stated as a natural transformation between functors of the same variance.
• Define equivalence of categories and prove that a functor is an equivalence if and only if it is full, faithful and essentially surjective.

Definition

• A natural transformation α : F ⇒ G between functors F, G : C → D consists of morphisms α_A : F(A) → G(A), one for each object A, such that G(f) ∘ α_A = α_B ∘ F(f) for every morphism f : A → B.
• A natural isomorphism is a natural transformation all of whose components are isomorphisms.
• An equivalence of categories is a functor F : C → D together with a functor G : D → C and natural isomorphisms G ∘ F ≅ id_C and F ∘ G ≅ id_D.

Theorem

• Functors C → D and the natural transformations between them form a functor category [C, D].
• A functor is an equivalence if and only if it is fully faithful and essentially surjective (the 'if' direction uses the axiom of choice in general).
• The category of finite-dimensional vector spaces over k is equivalent to its full subcategory on the spaces kⁿ, whose morphisms are matrices.

Common misconceptions

• Thinking 'natural' means 'obvious' or 'canonical-looking'; it is a precise commuting condition.
• Believing equivalent categories must be isomorphic.

Leads to

Optional discoveries

• Algebraic topology Eilenberg and Mac Lane introduced categories, functors and natural transformations (1945) to make precise the naturality of constructions in algebraic topology, such as homology.
• ← Categories
• ← Advanced linear algebra

Learning pathways

• Graduate preparation

Further reading

• Saunders Mac Lane, Categories for the Working Mathematician (1971)
• Tom Leinster, Basic Category Theory (2014)
• Emily Riehl, Category Theory in Context (2016)