Category Theory

Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and arrows (morphisms).

Categories and Morphisms

A category $\mathcal{C}$ consists of:

• A class $ob(\mathcal{C})$ of objects.
• A class $hom(\mathcal{C})$ of morphisms (arrows) between objects.
• A composition law $\circ$ that is associative and has identity morphisms.

Examples include Set (sets and functions), Grp (groups and homomorphisms), and Top (topological spaces and continuous maps).

class Category:
  def __init__(self, name):
      self.name = name
      self.objects = set()
      self.morphisms = {} # (A, B) -> list of f

  def add_object(self, obj):
      self.objects.add(obj)

  def add_morphism(self, source, target, name):
      if (source, target) not in self.morphisms:
          self.morphisms[(source, target)] = []
      self.morphisms[(source, target)].append(name)

# Modeling a simple Finite Category
C = Category("SimpleFinite")
C.add_object("A")
C.add_object("B")
C.add_morphism("A", "B", "f")
C.add_morphism("B", "B", "id_B")

print(f"Objects in {C.name}: {C.objects}")
print(f"Morphisms: {C.morphisms}")

Functors

A Functor $F: \mathcal{C} \to \mathcal{D}$ is a mapping that preserves the category structure:

1. It maps objects in $\mathcal{C}$ to objects in $\mathcal{D}$.
2. It maps morphisms $f: A \to B$ in $\mathcal{C}$ to morphisms $F(f): F(A) \to F(B)$ in $\mathcal{D}$.
3. It preserves identity and composition.

Natural Transformations

A natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. It is often described as a “morphism between functors.”

Yoneda Lemma

The Yoneda Lemma suggests that an object is entirely determined by its relationships (morphisms) with other objects in the category. This is a foundational result that allows one to embed any category into a category of functors.