Functor

The term functor has two distinct meanings.

In Pure Mathematics

In category theory, a functor is a map between categories satisfying certain relations. Functors, in a sense, provide for categories what group homomorphisms do for groups. To be precise, a function $F:{\mathcal {C}}\to {\mathcal {D}}$ between two categories associates to each object $X\in {\textrm {Ob}}{\mathcal {C}}$ an object $F(X)\in {\textrm {Ob}}{\mathcal {D}}$, and to each morphism $f\in {\mathcal {C}}(X,Y)$ a morphism $F(f)\in {\mathcal {D}}(FX,FY)$ such that:

$F({\textrm {id}}_{X})=id_{F(X)}\,$

$F(g\circ f)=F(g)\circ F(f)$.

Functors are the fundamental objects used to relate structures between different categories.

The traditional terminology for the above is a "covariant functor", to distinguish it from "contravariant functors", which reverse the direction of compositions, and instead satisfy

$F(g\circ f)=F(f)\circ F(g)$.

We will see an example of this below.

Examples

Algebraic topology was the first field in which the usefulness of the notion of a functor was recognized. A basic example is the fundamental group functor $\pi _{1}:{\textbf {Top}}*\to {\textbf {Grp}}$. The action on objects is defined by sending a topological space to its fundamental group $\pi _{1}(X)$. Recall that a map between two topological spaces $f:X\to Y$ induces a map $f_{*}:\pi _{1}(X)\to \pi (Y)$ by $f_{*}([\gamma ])=[f\circ \gamma ]$. Set $\pi _{1}(f)=f_{*}\,$ so defined. The functoriality of $\pi _{1}$ boils down to the fact that $F({\textrm {id}}_{X})={\textrm {id}}_{\pi _{1}(X)}$, that is, the identity map on a topological space induces the identity map on its fundamental group, together with the fact that $F(g\circ f)=F(g)\circ F(f)$, explained at fundamental group.

A simple example arises in the category of vector spaces. Fix a vector space $V$. We can define a functor ${\textrm {Hom}}(V,-):{\textrm {Vect}}_{k}\to {\textrm {Vect}}_{k}$ by

1. An object $A$ (that is a vector space) is sent to ${\textrm {Hom}}(V,A)$, the vector space of linear maps from $V$ to $A$. We can think of these linear maps as matrices.
2. A linear transformation $\phi :A\to B$ (i.e., an element of ${\textrm {Hom}}_{{\textrm {Vect}}_{k}}(A,B)$ is sent to a linear transformation ${\textrm {Hom}}(V,A)\to {\textrm {Hom}}(V,B)$ by setting ${\textrm {Hom}}(V,-)(\phi )=(f\mapsto \phi \circ f)$. I other words, the image of $\phi$ is supposed to be a linear transformation from ${\textrm {Hom}}(V,A)$ to ${\textrm {Hom}}(V,B)$: it is defined by sending a linear transformation $f:V\to A$ to $\phi \circ f:V\to B$.

We can similarly obtain a contravariant function ${\textrm {Hom}}(-,V)$ by

1. An object $A$ is sent to the vector space of linear maps ${\textrm {Hom}}(A,V)$.
2. A map $\phi \in {\textrm {Hom}}(A,B)$ is sent to a linear map ${\textrm {Hom}}(B,V)\to {\textrm {Hom}}(A,V)$ by $\phi \mapsto (f\mapsto f\circ \phi )$. Given a linear map from B to V, we get one from A to V by composing with $\phi$. Note that morphisms now compose in the opposite direction: that is why this is a contravariant functor.

In Computer Science

In computer science, the term "functor" is short for "function operator", also called a "function object". It is essentially a class method with no name. Many modern programming languages support this. For example, in C++ one can define a function object with the "operator()" notation, that is, a definition of what parentheses mean after an object.

 class C {
 public:
    int operator()(int i, int j) { return k+i*j; }
    int k;
    .....
  };
  
  .....
  C MyObject;
  MyObject.k = 3;
  int r = MyObject(4,5);
  .....

We have effectively defined a nameless method for class C. Instead of making a named invocation like "MyObject.func(4,5)", we just write "MyObject(4,5)".