Categories Functors

When to Use

Use this skill when working on categories-functors problems in category theory.

Decision Tree

1. Verify Category Axioms

   • Objects and morphisms (arrows) defined?
   • Identity morphism for each object: id_A: A -> A
   • Composition associative: (f . g) . h = f . (g . h)
   • Write Lean 4: theorem assoc : (f ≫ g) ≫ h = f ≫ (g ≫ h) := Category.assoc

2. Check Functor Properties

   • F: C -> D maps objects to objects, arrows to arrows
   • Preserves identity: F(id_A) = id_{F(A)}
   • Preserves composition: F(g . f) = F(g) . F(f)
   • Write Lean 4: theorem comp : F.map (g ≫ f) = F.map g ≫ F.map f := F.map_comp

3. Functor Types

   • Covariant: preserves arrow direction
   • Contravariant: reverses arrow direction
   • Faithful/Full: injective/surjective on Hom-sets
   • Equivalence: full, faithful, essentially surjective

4. Common Functors

   • Forgetful functor: forgets structure (e.g., Grp -> Set)
   • Free functor: left adjoint to forgetful
   • Hom functor: Hom(A, -) or Hom(-, B)
   • Power set functor: Set -> Set via X |-> P(X)

5. Verify with Lean 4

   • Compiler-in-the-loop: write proof, lake build checks
   • Mathlib has full category theory library
   • See: .claude/skills/lean4-functors/SKILL.md for exact syntax

Tool Commands

Lean4_Category

# Lean 4 with Mathlib: import CategoryTheory.Category.Basic

Lean4_Functor

# Lean 4: theorem map_comp (F : C ⥤ D) : F.map (g ≫ f) = F.map g ≫ F.map f := F.map_comp

Lean4_Build

lake build  # Compiler-in-the-loop verification

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.