Contra-, Co- and Bivariant

Contravariant functors

A contravariant functor is a functor that reverse composition. That is given \( \mathcal{C},\mathcal{D} \) as categories, we define a contravariant functor as

• Associates an object \( F(X) \in \mathcal{D} \) for each \( X \in \mathcal{C} \).
• For each \( \mathcal{C} \)-arrow \( f : A \rightarrow B \) we have a \( \mathcal{D} \)-arrow \( F(f) : F(B) \rightarrow F(A) \) such that
  • \( F(id_A) = id_{F(A)} \) for every object \( A \in \mathcal{C} \).
  • \( F(g \circ f) = F(f) \circ F(g) \) for every \( \mathcal{C} \)-arrows \( f : A \rightarrow B \) and \( g : B \rightarrow C \).

That is if we have the \( \mathcal{C} \)-arrow \( g \circ f : A \rightarrow C \), then we have the \( \mathcal{D} \)-arrow \( F(f) \circ F(g) : F(C) \rightarrow F(A) \).

Covariant functors

Covariant functors are just normal functors called so in order to distinguish them from contravariant ones.

Note that contravariant functors are covariant in the dual category. That is given the contravariant functor \( F : \mathcal{C} \rightarrow \mathcal{D} \), we can write is as a covariant given as \( F : \mathcal{C}^{op} \rightarrow \mathcal{D} \).

Bivariant

A bifunctor is a functor with a domain as a product category. Eg. $$ \mathcal{A} \times \mathcal{B} \rightarrow \mathcal{C} $$

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