It often arises that the values of a type are partitioned into a variety of classes, each classifying data with distinct internal structure. A simple example is provided by the type of points in the plane, which are classified according to whether they are represented in cartesian or polar form. Both are represented by a pair of real numbers, but in the cartesian case, these are the x and y coordinates of the point, whereas in the polar case, these are its distance r from the origin and its angle θ with the polar axis. A classified value is an object, or instance, of its class. The class determines the type of the classified data, the instance type of the class; the classified data itself is the instance data of the object.

Methods are functions that act on classified values. The behavior of a method is determined by the class of its argument. The method dispatches on the class of the argument. Because the selection is made at run-time, it is called dynamic dispatch. For example, the squared distance of a point from the origin is calculated differently according to whether the point is represented in cartesian or polar form. In the former case, the required distance is x2 + y2, whereas in the latter it is simply r2. Similarly, the quadrant of a cartesian point can be determined by examining the sign of its x and y coordinates, and the quadrant of a polar point can be calculated by taking the integral part of the angle θ divided by π/2.

Dynamic dispatch is often described in terms of a particular implementation strategy, which we will call the class-based organization. In this organization, each object is represented by a vector of methods specialized to the class of that object. We may equivalently use a method-based organization in which each method branches on the class of an object to determine its behavior. Regardless of the organization used, the fundamental idea is that (a) objects are classified and (b) methods dispatch on the class of an object. The class-based and method-based organizations are interchangeable and, in fact, related by a natural duality between sum and product types.We explain this symmetry by focusing first on the behavior of each method on each object, which is given by a dispatch matrix.