Natural Transformations
- 1 installs
- 3.9k repo stars
- Updated January 26, 2026
- parcadei/continuous-claude-v3
Guides Claude through category-theory natural transformations by verifying naturality squares commute via Lean 4.
About
A decision-tree skill for natural transformations, verifying naturality-square commutativity between functors. A developer uses it when they want Claude to prove naturality conditions in Lean 4.
- Naturality square commutativity check
- Lean 4 naturality theorem
Natural Transformations by the numbers
- 1 all-time installs (skills.sh)
- Ranked #1,803 of 2,064 Data Science & ML skills by installs in the Skillselion catalog
- Data as of Aug 5, 2026 (Skillselion catalog sync)
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| Installs | 1 |
|---|---|
| repo stars | ★ 3.9k |
| Last updated | January 26, 2026 |
| Repository | parcadei/continuous-claude-v3 ↗ |
What it does
Guides Claude through category-theory natural transformations by verifying naturality squares commute via Lean 4.
Files
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.mdfor exact syntax
Tool Commands
Lean4_Naturality
# Lean 4: theorem nat : η.app B ≫ G.map f = F.map f ≫ η.app A := η.naturalityLean4_Nat_Trans
# Lean 4: def η : F ⟶ G where app := fun X => component_XLean4_Yoneda
# Lean 4: CategoryTheory.yonedaEquiv -- Yoneda lemmaLean4_Build
lake build # Compiler-in-the-loop verificationCognitive Tools Reference
See .claude/skills/math-mode/SKILL.md for full tool documentation.
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