The idea of the universal property is fundamental to category theory. Category theory is famously very difficult to understand, even for people we would otherwise expect to be at Math 3; universal properties are perhaps the easiest important theme of category theory.

The purpose of this page is to sketch out what an Intro to Universal Properties project might look like, so we can evaluate whether it would make a good project.

Goal of the project

• (Ambitious) Provide a guide to universal properties that would be enlightening to Math 1 readers. This need not specifically “be about” category theory; we could approach it using (pictures of) partial orders as very simple examples of categories, and then using the union/​product of finite sets.

• Provide a guide to universal properties that would be enlightening to Math 2 and 3 readers.

• Create a framework that is extensible with more and more examples of universal properties, as variations on a theme.

Outline

1. The idea of “not caring about things except up to isomorphism”.

2. The idea that we can describe objects based entirely on how they interact with other objects.

3. Introduce the category of finite sets, describing the empty set, disjoint union and product

4. Show how the empty set can be described entirely by its universal property.

5. Show how the union and product can be described entirely by their universal properties, up to isomorphism.

6. Introduce a specific poset category: \(\mathbb{N}\) with an arrow between \(a\) and \(b\) iff \(a\) divides \(b\). (Not sure about this one—maybe it already requires knowing what a category is?)

7. Describe the least upper bound and greatest lower bounds.

8. Describe the universal properties of the LUB and GLB; compare them with the union and coproduct.

9. Wrap up by explaining that this kind of property crops up all over the place.