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 <math>F : \mathcal C \to \mathcal D</math> between two categories associates to each object <math>X \in \textrm{Ob } \mathcal C</math> an object <math>F(X) \in \textrm{Ob } \mathcal D</math>, and to each morphism <math>f \in \mathcal C(X,Y)</math> a morphism <math>F(f) \in \mathcal D(FX,FY)</math> such that:

<math>F(\textrm{id}_X) = id_{F(X)}\,</math>

<math>F(g \circ f) = F(g) \circ F(f)</math>.

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

<math>F(g \circ f) = F(f) \circ F(g)</math>.

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 <math>\pi_1 : \textbf{Top}* \to \textbf{Grp}</math>. The action on objects is defined by sending a topological space to its fundamental group <math>\pi_1(X)</math>. Recall that a map between two topological spaces <math>f : X \to Y</math> induces a map <math>f_* : \pi_1(X) \to \pi(Y)</math> by <math>f_*([\gamma]) = [f \circ \gamma]</math>. Set <math>\pi_1(f) = f_*\,</math> so defined. The functoriality of <math>\pi_1</math> boils down to the fact that <math>F(\textrm{id}_X) = \textrm{id}_{\pi_1(X)}</math>, that is, the identity map on a topological space induces the identity map on its fundamental group, together with the fact that <math>F(g \circ f) = F(g) \circ F(f)</math>, explained at fundamental group.

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

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

We can similarly obtain a contravariant function <math>\textrm{Hom}(-,V)</math> by

1. An object <math>A</math> is sent to the vector space of linear maps <math>\textrm{Hom}(A,V)</math>.
2. A map <math>\phi \in \textrm{Hom}(A,B)</math> is sent to a linear map <math>\textrm{Hom}(B,V) \to \textrm{Hom}(A,V)</math> by <math> \phi \mapsto (f \mapsto f \circ \phi)</math>. Given a linear map from B to V, we get one from A to V by composing with <math>\phi</math>. 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)".