Natural Transformations

Natural Transformations

When to Use

Use this skill when working on natural-transformations problems in category theory.

Decision Tree

1. Verify Naturality

   • eta: F => G is natural transformation between functors F, G: C -> D
   • For each f: A -> B in C, diagram commutes: G(f) . eta_A = eta_B . F(f)
   • Write Lean 4: theorem nat : η.app B ≫ G.map f = F.map f ≫ η.app A := η.naturality

2. Component Analysis

   • eta_A: F(A) -> G(A) for each object A
   • Each component is morphism in target category D
   • Lean 4: def η : F ⟶ G where app := fun X => ...

3. Natural Isomorphism

   • Each component eta_A is isomorphism
   • Functors F and G are naturally isomorphic
   • Notation: F ≅ G (NatIso in Mathlib)

4. Functor Category

   • [C, D] has functors as objects
   • Natural transformations as morphisms
   • Vertical composition: Lean 4 CategoryTheory.NatTrans.vcomp
   • Horizontal composition: CategoryTheory.NatTrans.hcomp

5. Yoneda Lemma Application

   • Nat(Hom(A, -), F) ~ F(A) naturally in A
   • Lean 4: CategoryTheory.yonedaEquiv
   • Fully embeds C into [C^op, Set]
   • See: .maestro/skills/lean4-nat-trans/SKILL.md for exact syntax

Tool Commands

Lean4_Naturality

# Lean 4: theorem nat : η.app B ≫ G.map f = F.map f ≫ η.app A := η.naturality

Lean4_Nat_Trans

# Lean 4: def η : F ⟶ G where app := fun X => component_X

Lean4_Yoneda

# Lean 4: CategoryTheory.yonedaEquiv -- Yoneda lemma

Lean4_Build

lake build  # Compiler-in-the-loop verification

Cognitive Tools Reference

See .maestro/skills/math-mode/SKILL.md for full tool documentation.