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.

  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

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

  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.