In category theory, there's the idea of the product as an object satisfying a particular universal property. Can you suggest ways to make the concept of the product intuitive? (So far, my attempt contains three examples: "product of finite sets" and two instances of "product in a poset category", and I'm teaching primarily by example.)
I've recently been writing a series about the category-theoretic concept of the universal property, over on Arbital (an open collaborative maths-explanations site). The aim is to make an intuitive explanation for people who are comfortable enough with mathematical notation that they can read formulae if they have explanations attached, and who are completely comfortable with the notion of a "function", but who have no exposure to category theory. (For example, fairly advanced students of computer science or engineering, or second-year undergraduate mathematics students. I studied maths, so I know the level of mathematics study I want from the students; my guesses for compsci and engineering are less well-founded.)
I'm running into what I presume are two of the standard problems (certainly the nLab has both these problems in spades, because it's not really designed to teach, but to be a reference work):
- I don't really know whether my explanation would teach someone who didn't already know the content. The only person we can get to dog-food this content is someone who already kind of knows the shape of some basic category theory.
- Category theory turns out to be quite difficult to explain. (Who knew.)
We've already got Universal Property of the Empty Set, which I'm pretty pleased with; now I'm trying to make intuitive the notion of the product, on Universal Property of the Product. The latter is what this Question is about.
The following is not necessary for an answer to this Question, but if you have any suggested specific improvements to the Product page, I'd be very happy to hear them. Similarly, anyone assisting in any way with the rest of the Universal Properties project would receive twenty Internet points from me.