Real Numbers

example

The real numbers $\mathbb{R}$ form a Preorder with the “usual ordering”, e.g. $-500\le -499\le 0\le \sqrt{2}\le 100/3$. It is a Total Order (7S Exercise 1.48) and a Partial Order.

Sources: 7 Sketches Example 1.47, 1.97, §1.3.1; Kittenlab Lecture 5, 7, 10, 11.

• The subset $\mathbb{N}\subseteq \mathbb{R}$ has a Meet ($0$) but no Join; $\{\frac{1}{n+1}\mid n\in \mathbb{N}\}$ has meet $0$ and join $1$ (7S Exercise 1.80).
• The maps $\lceil -/3\rceil ⊣(3\times -)⊣\lfloor -/3\rfloor$ between $\mathbb{Z}$ and $\mathbb{R}$ form Galois connections (Example 1.97, 7S Exercise 1.98).
• 7S Exercise 1.1 distinguishes order-, metric-, and addition-preserving functions $\mathbb{R}\to \mathbb{R}$.
• Kittenlab: order-preserving maps $\mathbb{R}\to \mathbb{R}$ form a poset under pointwise order, the natural transformations between them (Lecture 7); the functor $U:\mathbf{Vect}\to \mathbf{Set}$ is represented by $\mathbb{R}$ (Lecture 10); the Coequalizer of $\mathrm{id},(+2\pi ):\mathbb{R}\rightrightarrows \mathbb{R}$ is the circle (Lecture 11); in $({\mathbb{Q}}_{\ge 0},\le )$ the functor $[2\le x^2]$ is not representable but in ${\mathbb{R}}_{\ge 0}$ it is, represented by $\sqrt{2}$.
• $[0,\infty ]$ with the reverse order is the base of enrichment for Lawvere metric spaces (Cost).