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