Explaining Category Theory With Object-Oriented Programming

I recently came across Category Theory for the Java Programmer. At various points in the Poincaré Project, I've considered writing code to illustrate mathematical structures. Recently reading about category theory (and topoi in particular) I have myself been tempted to think of them in terms of OO programming.

My inclination would, of course, be to use Python, but static typing is probably more useful for explicitly showing the structure of mathematical objects (although at the end of the day, mathematics is about duck typing: if it follows the axioms of a linear space, it is a linear space).

So the post is pretty cool. But what I felt got in the way of using Java to explain category theory is that category theory is essentially about relationships between classes (in the OO sense—or least what OOA/OOD thinks of as what classes are supposed to map to in the real world). The objects in a category like Set are sets, not elements. The objects in a category like Vect are vector spaces, not vectors. Now one might argue that category theory doesn't really care about elements and vectors, only sets and vector spaces. So you could just shift your "metaness" one to the left. One might also argue (as the blog post referenced at the start does) that classes in Java are instances of the class Class so calling a class an object is allowed. But I think it just makes things confusing for the average OO programmer (especially if you think OO == Java).

A better language for illustrating category theory would be one that has a more explicit notion of a metaclass. Then it would be easier to explain category theory as essentially about different characteristics of metaclasses.

The original post was in the category: mathematics but I'm still in the process of migrating categories over.