# Definition:Functor/Contravariant

## 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 $\mathbf C$:

$\map F {g \circ f} = F f \circ F g$

and:

$\map F {\operatorname {id}_X} = \operatorname {id}_{F X}$

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 $\mathbf C$.

## Co- and Contravariance

Both covariant and contravariant functors are paramount in all of contemporary mathematics.

The intention behind defining a functor is to formalise and abstract the intuitive notion of "preserving structure".

Functors thus can be understood as a generalisation of the concept of homomorphism in all its instances.

This explains why one would be led to contemplate covariant functors.

However, certain "natural" operations like transposing a matrix do not preserve the structure as rigidly as a homomorphism (we do have Transpose of Matrix Product, however).

Because of the abundant nature of this type of operation, the concept of a contravariant functors was invented to capture their behaviour as well.