The Theory of Classification, Part 7: A Class is a Type Family

This is the seventh article in a regular series on object-oriented type theory, aimed specifically at non-theoreticians. So far, we have built up a model of objects as simple records, which are instances of corresponding record types [1]. Initially, we took the seemingly attractive view that a programmer's class in C++ or Java corresponds in some way to a type in the formal model, and that a compatible subclass therefore corresponds to a subtype [2], linking this with an algebraic description of subtype-compatible semantic behaviour [3]. Naturally, the class-construct in an object-oriented language also defines an implementation for the instances of the class, but there is no formal contradiction here in considering just the typeful aspects of class declarations. Subtyping does indeed provide a simple, flexible model for type compatibility, but we shall find that it is not always as useful as we might expect.