Project outline: Intro to the Universal Property

Category theory is famously very difficult to understand, even for people with a relatively high level of mathematical maturity. Universal properties are perhaps the easiest important theme of category theory.

With this project, we want to produce an explanation that will clearly communicate this core concept in category theory, the universal property, to a wide audience of learners.

This page is an outline for the project, the below links are to pages within its scope.

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

• Isomorphism

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

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

• Set

• Finite set

• Empty set

• Union

• Disjoint union of sets

• Set product

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

• Universal property

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

2. 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?)

3. Describe the least upper bound and greatest lower bounds in a poset; in particular, in \(\mathbb{N}\) under the divisibility relation, we obtain the GCD and the LCM.

• A page (or two) about poset least upper bound and greatest lower bound in a poset (these actually already exist! join and meet)

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

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