functor

Examples:

• If X is almost any category, it has a "forgetful functor" which copies every node A to the set A of its elements. For example, this lets you treat groups as just the sets of their elements, forgetting the special group structure defined on them. This functor is the basis of rp's assertion in his writeup under category theory that "nodes are sets". They're not, but we can usually think of them as sets by forgetting about their extra properties.
• If A is a category and A' a pointed category based on A, there is a slightly less forgetful functor from A' to A which takes (A,a) to A (it "forgets" about a).
• The category Group of groups has a nice functor from Group to Group which takes every group G to a group with exactly the same elements, but the operation takes the elements in opposite order. The groups which this functor takes to themselves are exactly the category of Abelian groups.

In C++ a functor is the name of a special kind of operator overloading. Defining a functor for a class allows you to treat an object as if it were a function. This can be confusing, so is rarely seen. An example would be:

class CSquare
{
public:
    CSquare(){}
    int operator()( int i )
    { return( i * i ); }
};

Then, using the above class definition, you could write something like the following:

int main()
{
    CSquare square;

    int s = square( 5 );
}

One possible use for functors is for multiple subscript arrays (i.e. matrices). Since overloading the bracket operator only allows a single parameter inside the brackets, you could instead define a functor that taks multiple subscripts.

typedef void (function_t)(); // Type of a function taking no args and
                             // returning void

int functor(function_t * const arg); // direct declaration of a function
                                     // that is a functor