natural-transformations skill
Problem-solving strategies for natural transformations in category theory
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Install the natural-transformations skill
A skill is a folder. Copy it into your agent's skills folder and the agent loads it when the task matches its description.
git clone --depth 1 https://github.com/parcadei/Continuous-Claude-v3.git /tmp/Continuous-Claude-v3 mkdir -p ~/.claude/skills cp -r /tmp/Continuous-Claude-v3/.claude/skills/math/category-theory/natural-transformations ~/.claude/skills/natural-transformations
available in every project
In the Claude apps, zip the folder and upload it from the Skills settings. The folder on GitHub
The instructions your agent would load
SKILL.md as published, without the frontmatter. Read it on GitHub
Natural Transformations
When to Use
Use this skill when working on natural-transformations problems in category theory.
Decision Tree
- Verify Naturality
G(f) . etaA = etaB . F(f)
- eta: F => G is natural transformation between functors F, G: C -> D
- For each f: A -> B in C, diagram commutes:
- Write Lean 4: theorem nat : η.app B ≫ G.map f = F.map f ≫ η.app A := η.naturality
- 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 => ...
- Natural Isomorphism
- Each component eta_A is isomorphism
- Functors F and G are naturally isomorphic
- Notation: F ≅ G (NatIso in Mathlib)
- Functor Category
- [C, D] has functors as objects
- Natural transformations as morphisms
- Vertical composition: Lean 4 CategoryTheory.NatTrans.vcomp
- Horizontal composition: CategoryTheory.NatTrans.hcomp
- 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.naturalityLean4NatTrans
# 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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