Universal property of joins and meets in a poset


by Patrick Stevens Dec 31 2016

Partially ordered sets are a good setting to illustrate some of the concepts of category theory; joins and meets are instances of category-theoretic ideas.

[summary(Technical): In a Partially ordered set, the Join and meet are instances of the category-theoretic product and coproduct.]

[summary: We can see some of the ideas of Category theory by looking at the very simple examples of categories, such as Partially ordered sets. In this setting, the Join and meet turn out to correspond to the product and coproduct.]

[todo: state the UP of the product, using the divisibility in $~$\mathbb{N}$~$ example and the example of subsets of $~$\{1,2,3,4,5\}$~$] [todo: state the UP of the coproduct, showing that it's the same as the product in $~$P^{\text{op}}$~$] [todo: relate the product and coproduct UPs to the general concepts]