Project proposal: Intro to the Universal Property

by Patrick Stevens Aug 16 2016

Proposal for one of the first Arbital Projects.

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


  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.


Eric Rogstad

We're going to feature whatever we choose as the current project on the front page, and I want to include some intro text. What do you think of the following (adapted from the first paragraph above):

Help us build an intuitive explanation of this fundamental concept in category theory!

Category theory is famously very difficult to understand, even for people with a relatively high level of mathematical maturity. With this project, we want to produce an explanation that will clearly communicate a core concept in category theory, the universal property, to a wide audience of learners.

See below for our current progress on the project, as well as how you can contribute.