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

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.

Parents: