/natural-transformations

name: natural-transformations
description: "Problem-solving strategies for natural transformations in category theory"
allowed-tools: [Bash, Read]

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: `.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.