Definition:Functor/Contravariant

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Definition

Definition 1

Let $\mathbf C$ and $\mathbf D$ be metacategories.


A contravariant functor $F : \mathbf C \to \mathbf D$ consists of:

  • An object functor $F_0$ that assigns to each object $X$ of $\mathbf C$ an object $FX$ of $\mathbf D$.
  • An arrow functor $F_1$ that assigns to each arrow $f : X \to Y$ of $\mathbf C$ an arrow $Ff : FY \to FX$ of $\mathbf D$.


These functors must satisfy, for any morphisms $X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z$ in $\mathbf C$:

$F \left({g \circ f}\right) = Ff \circ Fg$

and:

$F \left({\operatorname{id}_X}\right) = \operatorname{id}_{FX}$

where:

$\operatorname{id}_W$ denotes the identity arrow on an object $W$

and:

$\circ$ is the composition of morphisms.


Definition 2

Let $\mathbf C$ and $\mathbf D$ be metacategories.

A contravariant functor $F : \mathbf C \to \mathbf D$ is a covariant functor:

$F: \mathbf C^{\text{op}} \to \mathbf D$

where $\mathbf C^{\text{op}}$ is the dual category of $\mathbf C$.


Let $\mathbf C$ and $\mathbf D$ be metacategories.

A contravariant functor $F : \mathbf C \to \mathbf D$ is a covariant functor $F: \mathbf C^{\text{op}} \to \mathbf D$, where $\mathbf C^{\text{op}}$ is the dual category of $\mathbf C$.


Equivalence of Definitions

The definitions above are equivalent.

This is shown on Equivalence of Definitions of Contravariant Functor.


Also see