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.
where $\operatorname {id}_W$ denotes the identity arrow on an object $W$, and $\circ$ is the composition of morphisms.
The behaviour of a contravariant functor can be pictured as follows:
- $\begin{xy} <4em,4em>*{\mathbf C} = "C", <0em,0em>*+{X} = "a", <4em,0em>*+{Y} = "b", <4em,-4em>*+{Z}= "c", "a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, "b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g}, "a";"c" **@{-} ?>*@{>} ?<>(.5)*!/r1.5em/{g \circ f}, "C"+/r9em/*{\mathbf D}, "C"+/r2em/;"C"+/r6em/ **@{-} ?>*@{>} ?*!/_1em/{F}, "b"+/r2em/+/_2em/;"b"+/r6em/+/_2em/ **@{~} ?>*@2{>} ?<>(.5)*!/_.6em/{F}, "a"+/r13em/*+{FX}="Fa", "b"+/r13em/*+{FY}="Fb", "c"+/r13em/*+{FZ}="Fc", "Fb";"Fa" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{Ff}, "Fc";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/^1em/{Fg}, "Fc";"Fa" **@{-} ?>*@{>} ?<>(.7)*!/_1.5em/{\map F {g \circ f}}, \end{xy}$
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.