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 $\mathbf C$.
Also see
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.