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(0,1)-category

(0,1)-topos

# Contents

## Idea

The enriched generalization of the notion of monotone function.

## Definition

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 $y \in A$.

## Examples

• 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 $B$.