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

1. 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 => ...

1. Natural Isomorphism

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

1. Functor Category

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

1. Yoneda Lemma Application

• Nat(Hom(A, -), F) ~ F(A) naturally in A
• Lean 4: CategoryTheory.yonedaEquiv
• Fully embeds C into [C^op, Set]
• See: .claude/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 .claude/skills/math-mode/SKILL.md for full tool documentation.