vibecosystem

Natural Transformations

When to Use

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

Decision Tree

1. Verify Naturality
2. eta: F => G is natural transformation between functors F, G: C -> D
3. 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
2. eta_A: F(A) -> G(A) for each object A
3. Each component is morphism in target category D
4. Lean 4: def η : F ⟶ G where app := fun X => ...

1. Natural Isomorphism
2. Each component eta_A is isomorphism
3. Functors F and G are naturally isomorphic
4. Notation: F ≅ G (NatIso in Mathlib)

1. Functor Category
2. [C, D] has functors as objects
3. Natural transformations as morphisms
4. Vertical composition: Lean 4 CategoryTheory.NatTrans.vcomp
5. Horizontal composition: CategoryTheory.NatTrans.hcomp

1. Yoneda Lemma Application
2. Nat(Hom(A, -), F) ~ F(A) naturally in A
3. Lean 4: CategoryTheory.yonedaEquiv
4. Fully embeds C into [C^op, Set]
5. 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.