1.2 Basic definitions and notation

Category theory is a powerful framework for understanding mathematical structures and relationships. It focuses on objects and morphisms, providing a unified language for describing diverse mathematical concepts across different fields.

At its core, category theory deals with composition, identity, and associativity. These fundamental principles allow us to analyze complex systems by breaking them down into simpler components and understanding how they interact.

Fundamental Concepts of Category Theory

Components of categories

• A category $\mathcal{C}$ consists of two main components:
  • Objects denoted as $A$, $B$, $C$, etc. represent the entities or structures within the category (groups, sets, vector spaces)
  • Morphisms (arrows) denoted as $f$, $g$, $h$, etc. represent the relationships, transformations, or mappings between objects (group homomorphisms, functions, linear transformations)
• For every pair of objects $A$ and $B$, there exists a set $\text{Hom}_{\mathcal{C}}(A, B)$ called the hom-set
  • Contains all morphisms from object $A$ to object $B$, capturing the possible ways to relate or transform $A$ into $B$
  • If $f \in \text{Hom}_{\mathcal{C}}(A, B)$, it is denoted as $f: A \to B$, indicating that $f$ is a morphism from $A$ to $B$ (function from set $A$ to set $B$)

Notation for category theory

• Composition of morphisms: If $f: A \to B$ and $g: B \to C$, then there exists a morphism $g \circ f: A \to C$
  • The composition is read from right to left: $(g \circ f)(a) = g(f(a))$ for $a \in A$, meaning first apply $f$ to $a$, then apply $g$ to the result
  • Represents the idea of chaining transformations or mappings together (composing functions, group homomorphisms, or linear transformations)
• Commutative diagrams used to represent the equality of compositions of morphisms
  • If $f: A \to B$, $g: B \to C$, $h: A \to C$, and $h = g \circ f$, then the following diagram commutes: A @>f>> B \\ @VhVV @VVgV \\ C @= C \end{CD}$$
  • Visually captures the idea that following different paths in the diagram yields the same result (going from $A$ to $C$ directly via $h$ is the same as going from $A$ to $B$ via $f$, then from $B$ to $C$ via $g$)

Properties of morphisms

• Identity morphisms: For every object $A$, there exists a unique morphism $\text{id}_A: A \to A$ called the identity morphism
  • For any morphism $f: A \to B$, the following hold:
    • $f \circ \text{id}_A = f$, meaning composing $f$ with the identity on its domain yields $f$ itself (identity function, identity matrix)
    • $\text{id}_B \circ f = f$, meaning composing $f$ with the identity on its codomain yields $f$ itself
  • Captures the idea of a neutral element or transformation that preserves the object
• Associativity: For morphisms $f: A \to B$, $g: B \to C$, and $h: C \to D$, the following equality holds:
  • $(h \circ g) \circ f = h \circ (g \circ f)$, meaning the order of composition does not matter when grouping morphisms (associativity of function composition, matrix multiplication)
  • Allows for unambiguous composition of multiple morphisms without the need for parentheses

Types of morphisms

• Isomorphisms: A morphism $f: A \to B$ is an isomorphism if there exists a morphism $g: B \to A$ such that:
  • $g \circ f = \text{id}_A$ and $f \circ g = \text{id}_B$, meaning $f$ and $g$ are inverses of each other
  • If an isomorphism exists between $A$ and $B$, they are considered isomorphic, denoted as $A \cong B$, indicating that $A$ and $B$ are essentially the same object up to relabeling (isomorphic groups, homeomorphic spaces)
• Monomorphisms: A morphism $f: A \to B$ is a monomorphism (monic) if for any object $C$ and morphisms $g_1, g_2: C \to A$, the following implication holds:
  • If $f \circ g_1 = f \circ g_2$, then $g_1 = g_2$, meaning $f$ is left-cancellative or injective (injective functions, injective group homomorphisms)
• Epimorphisms: A morphism $f: A \to B$ is an epimorphism (epic) if for any object $C$ and morphisms $h_1, h_2: B \to C$, the following implication holds:
  • If $h_1 \circ f = h_2 \circ f$, then $h_1 = h_2$, meaning $f$ is right-cancellative or surjective (surjective functions, surjective group homomorphisms)