12.2 Contravariant functors

Contravariant functors flip the direction of morphisms between categories, unlike their covariant counterparts. They're essential in category theory, offering unique perspectives on relationships between mathematical structures.

Understanding contravariant functors helps grasp the duality principle in mathematics. They're used in various fields, from linear algebra to set theory, providing powerful tools for analyzing and transforming mathematical objects.

Contravariant Functors

Contravariant functors and applications

• Contravariant functors reverse the direction of morphisms in a category
  • Given categories $C$ and $D$, a contravariant functor $F: C \to D$ maps objects of $C$ to objects of $D$
  • Morphisms $f: A \to B$ in $C$ are mapped to morphisms $F(f): F(B) \to F(A)$ in $D$, effectively reversing the direction of the arrows
• Contravariant functors preserve composition and identity morphisms
  • For morphisms $f: A \to B$ and $g: B \to C$ in $C$, the composition is preserved: $F(g \circ f) = F(f) \circ F(g)$
  • For an object $A$ in $C$, the identity morphism is preserved: $F(id_A) = id_{F(A)}$
• Examples of contravariant functors showcase their utility in various mathematical contexts
  • The dual vector space functor $(\cdot)^*: Vect_k \to Vect_k$ maps a vector space $V$ to its dual space $V^*$ and a linear transformation $f: V \to W$ to its dual $f^*: W^* \to V^*$
  • The powerset functor $\mathcal{P}: Set^{op} \to Set$ maps a set $X$ to its powerset $\mathcal{P}(X)$ and a function $f: X \to Y$ to the preimage map $f^{-1}: \mathcal{P}(Y) \to \mathcal{P}(X)$

Covariant vs contravariant functors

• Covariant functors preserve the direction of morphisms between categories
  • Given categories $C$ and $D$, a covariant functor $F: C \to D$ maps objects of $C$ to objects of $D$
  • Morphisms $f: A \to B$ in $C$ are mapped to morphisms $F(f): F(A) \to F(B)$ in $D$, preserving the direction of the arrows
• Contravariant functors reverse the direction of morphisms between categories
  • Given categories $C$ and $D$, a contravariant functor $F: C \to D$ maps objects of $C$ to objects of $D$
  • Morphisms $f: A \to B$ in $C$ are mapped to morphisms $F(f): F(B) \to F(A)$ in $D$, reversing the direction of the arrows
• Both covariant and contravariant functors preserve composition and identity morphisms, ensuring the structure of the categories is maintained

Composition of contravariant functors

• The composition of two contravariant functors results in a covariant functor
• Let $F: C \to D$ and $G: D \to E$ be two contravariant functors
• The composition $G \circ F: C \to E$ is a covariant functor, as demonstrated by the following proof:
  1. For objects $A$ in $C$, $(G \circ F)(A) = G(F(A))$
  2. For morphisms $f: A \to B$ in $C$:
     • $F(f): F(B) \to F(A)$ in $D$ (contravariant)
     • $G(F(f)): G(F(A)) \to G(F(B))$ in $E$ (contravariant)
     • $(G \circ F)(f) = G(F(f)): (G \circ F)(A) \to (G \circ F)(B)$ (covariant)
  3. Composition is preserved:
     • For morphisms $f: A \to B$ and $g: B \to C$ in $C$: $(G \circ F)(g \circ f) = G(F(g \circ f)) = G(F(f) \circ F(g)) = G(F(f)) \circ G(F(g)) = (G \circ F)(f) \circ (G \circ F)(g)$
  4. Identity morphisms are preserved:
     • For an object $A$ in $C$, $(G \circ F)(id_A) = G(F(id_A)) = G(id_{F(A)}) = id_{G(F(A))} = id_{(G \circ F)(A)}$

Opposite Categories and Contravariant Functors

Contravariant from covariant functors

• The opposite category $C^{op}$ of a category $C$ is defined as follows:
  • $Obj(C^{op}) = Obj(C)$, meaning the objects of $C^{op}$ are the same as the objects of $C$
  • For objects $A, B$ in $C^{op}$, $Hom_{C^{op}}(A, B) = Hom_C(B, A)$, meaning the morphisms in $C^{op}$ are the morphisms in $C$ with reversed direction
  • Composition in $C^{op}$ is defined as $g \circ^{op} f = f \circ g$ for morphisms $f: B \to A$ and $g: C \to B$ in $C$
• Given a covariant functor $F: C \to D$, we can obtain a contravariant functor $F^{op}: C^{op} \to D$ as follows:
  1. For objects $A$ in $C^{op}$, $F^{op}(A) = F(A)$
  2. For morphisms $f: A \to B$ in $C^{op}$ (which correspond to morphisms $f: B \to A$ in $C$), $F^{op}(f) = F(f)$
• Proof that $F^{op}$ is a contravariant functor:
  1. For morphisms $f: A \to B$ and $g: B \to C$ in $C^{op}$ (which correspond to morphisms $f: B \to A$ and $g: C \to B$ in $C$): $F^{op}(g \circ^{op} f) = F^{op}(f \circ g) = F(f \circ g) = F(g) \circ F(f) = F^{op}(f) \circ F^{op}(g)$
  2. For an object $A$ in $C^{op}$, $F^{op}(id_A) = F(id_A) = id_{F(A)} = id_{F^{op}(A)}$