natural-transformationslisted

# 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 ```bash # Lean 4: theorem nat : η.app B ≫ G.map f = F.map f ≫ η.app A := η.naturality ``` ### Lean4_Nat_Trans ```bash # Lean 4: def η : F ⟶ G where app := fun X => component_X ``` ### Lean4_Yoneda ```bash # Lean 4: CategoryTheory.yonedaEquiv -- Yoneda