Project outline: Intro to the Universal Property

This page is an outline for the Universal Property project.

1. The idea of not caring about things except up to isomorphism.
   • 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
   • Set
   • Finite_set
   • Empty_set
   • Union
   • Set_disjoint_union
   • Set_product
4. Show how the empty set can be described entirely by its universal property.
   • 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: $N$ 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 in a poset; in particular, in $N$ under the divisibility relation, we obtain the GCD and the LCM.
   • A page (or two) about least_upper_bound and greatest_lower_bound
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.