Structure-preserving maps between categories
A functor is a bridge between two categories: it carries every dot and every arrow across while keeping all the connections intact.
Press Run Functor and watch each object and arrow travel from C into D. Notice that composites land on composites.
If categories are "worlds of mathematical objects," then functors are the bridges between them. A functor maps objects to objects and arrows to arrows, preserving all the structure. They are the morphisms in the "category of categories."
A functor F : C → D is a "machine" that takes an entire category and maps it into another, preserving all the structure: objects map to objects, arrows map to arrows, and composition is respected.
Key insight: A functor must satisfy: F(g ∘ f) = F(g) ∘ F(f) and F(id_A) = id_{F(A)}. It's a "homomorphism of categories" — it preserves all the categorical structure.
Switch between covariant and contravariant and watch the arrows keep or flip their direction.
A covariant functor preserves arrow direction. A contravariant functor reverses arrows — it's a functor from Cop to D.
Key insight: Contravariant functors are surprisingly common: the dual space functor V → V* reverses arrows, and so does the power set functor via preimage. A contravariant functor C → D is the same as a covariant functor Cop → D.
Click the button to forget the group operation and watch the structure dissolve, leaving only the underlying set.
| · | e | a | b | ab |
|---|---|---|---|---|
| e | e | a | b | ab |
| a | a | e | ab | b |
| b | b | ab | e | a |
| ab | ab | b | a | e |
The forgetful functor U : Grp → Set takes a group and "forgets" the group operation, leaving just the underlying set of elements. Click to see the structure dissolve!
Key insight: Forgetful functors "strip away" algebraic structure. The functor U : Grp → Set forgets the operation, U : Top → Set forgets the topology, and U : Vect → Set forgets the vector space structure. Forgetful functors almost always have a left adjoint (the "free" construction) — a deep fact we'll explore later!
Toggle between Hom(A, −) and Hom(−, A) and watch which way the arrows point.
Fix an object A. The covariant hom-functor Hom(A, −) sends each object B to the set of arrows from A to B. The contravariant version Hom(−, A) reverses arrow direction.
Key insight: The hom-functors are the most fundamental functors in category theory. They encode how an object relates to the rest of the category. The Yoneda Lemma (coming soon!) says that an object is completely determined by its hom-functor.
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