Set, Grp, Top, Vect, and poset categories in action
The power of category theory lies in its universality. The same framework describes wildly different areas of mathematics. Let's tour the most important categories and see how each branch of math fits into the categorical mold.
The most important category: Set has sets as objects and functions as morphisms. Every element in the domain maps to exactly one element in the codomain.
Key insight: In Set, bijections are exactly the isomorphisms — morphisms with a two-sided inverse. Composition is ordinary function composition, and identities are identity functions.
In Grp, objects are groups and morphisms are group homomorphisms — functions that preserve the group operation.
The inclusion Z₂ → Z₄ sends 0↦0 and 1↦2. This preserves the group operation: φ(a+b) = φ(a)+φ(b).
Key insight: A homomorphism must satisfy phi(a * b) = phi(a) * phi(b) — it preserves the group operation. The kernel (elements mapping to the identity) measures how much structure the homomorphism "forgets."
In Top, objects are topological spaces and morphisms are continuous maps. A continuous map never "tears" the space — nearby points stay nearby.
Key insight: In Top, the isomorphisms are homeomorphisms — continuous maps with continuous inverses. A coffee cup and donut are isomorphic in Top! The category structure captures exactly which properties are "topological."
In Vect, objects are vector spaces and morphisms are linear maps (matrices). Composition is matrix multiplication!
Key insight: In Vect, every morphism is a matrix. Composition is matrix multiplication, and isomorphisms are invertible matrices (det is not 0). Linear algebra is literally the study of one category!
Every partially ordered set (poset) is a category! Objects are elements, and there is a unique morphism a → b whenever a ≤ b. The Hasse diagram shows only the "covering" relations.
Key insight: A poset-as-category has at most one morphism between any two objects (a ≤ b gives a unique arrow). Reflexivity gives identity morphisms, and transitivity gives composition. This is the simplest non-trivial class of categories!