categories-functorslisted

# 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 ```bash # Lean 4 with Mathlib: import CategoryTheory.Category.Basic ``` ### Lean4_Functor ```bash # Lean 4: theorem map_comp (F : C ⥤ D) : F.map