4.1 Definition and types of functors

Functors are the backbone of category theory, mapping objects and morphisms between categories while preserving structure. They come in two flavors: covariant, which keep the direction of morphisms, and contravariant, which flip them.

Examples of functors include the identity functor, power set functor, and forgetful functor. Functors preserve composition and identity morphisms, ensuring the structure of the source category is maintained in the target category.

Functors

Functors as structure-preserving maps

• Functors map objects to objects and morphisms to morphisms between categories while preserving the categorical structure
• For categories $C$ and $D$, a functor $F: C \to D$ consists of:
  • Object function assigns each object $X$ in $C$ to an object $F(X)$ in $D$
  • Morphism function assigns each morphism $f: X \to Y$ in $C$ to a morphism $F(f): F(X) \to F(Y)$ in $D$
• Functors preserve the composition of morphisms
  • For morphisms $f: X \to Y$ and $g: Y \to Z$ in $C$, $F(g \circ f) = F(g) \circ F(f)$
• Functors preserve the identity morphisms
  • For any object $X$ in $C$, $F(id_X) = id_{F(X)}$

Covariant vs contravariant functors

• Covariant functors preserve the direction of morphisms
  • For $f: X \to Y$ in $C$, covariant functor $F$ maps it to $F(f): F(X) \to F(Y)$ in $D$
• Contravariant functors reverse the direction of morphisms
  • For $f: X \to Y$ in $C$, contravariant functor $F$ maps it to $F(f): F(Y) \to F(X)$ in $D$
  • Contravariant functors still preserve composition, but in the opposite order
    • For $f: X \to Y$ and $g: Y \to Z$ in $C$, $F(g \circ f) = F(f) \circ F(g)$

Examples of category functors

• Identity functor $1_C: C \to C$ maps each object and morphism in $C$ to itself
• Power set functor $\mathcal{P}: \mathbf{Set} \to \mathbf{Set}$ maps each set to its power set and each function to its induced function between power sets
• Dual functor $(\cdot)^{op}: C \to C^{op}$ maps each object in $C$ to itself and each morphism $f: X \to Y$ to its opposite $f^{op}: Y \to X$
• Forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$ maps each group to its underlying set and each group homomorphism to its underlying function

Preservation of morphism properties

• Given functor $F: C \to D$ and morphisms $f: X \to Y$, $g: Y \to Z$ in $C$:
  1. $F(g \circ f) = F(g) \circ F(f)$, demonstrating functors preserve composition
• For any object $X$ in $C$:
  • $F(id_X) = id_{F(X)}$, showing functors preserve identity morphisms
• These properties ensure the structure of the source category is maintained in the target category under the functor mapping