nLab enriched monotone

Contents

Idea

The enriched generalization of the notion of monotone function.

Given two preorders $A, B$ enriched in a monoidal poset $M$ , an $M$ -enriched monotone is a function $f: A \to B$ such that

o_A(x, y) \leq o_B(f(x), f(y))

for all $x \in A$ and $y \in A$ .

Let $M$ be $\Omega$ , the set of truth values, so that $A$ and $B$ are preorders. The $\Omega$ -enriched monotones between $A$ and $B$ are monotones between $A$ and $B$ .
Let $M$ be $(\mathbb{R}_{\geq 0}, \geq, +, 0)$ , the non-negative Dedekind real numbers with the preorder relation being greater than $\geq$ , the monoidal operation being addition $+$ , and the monoidal unit being zero $0$ , so that $A$ and $B$ are quasipseudometric spaces. The $\mathbb{R}_{\geq 0}$ -enriched monotones between $A$ and $B$ are distance-decreasing maps between $A$ and $B$ .