nLab enriched monotone

Contents

Contents

Idea

The enriched generalization of the notion of monotone function.

Definition

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

o A(x,y)o B(f(x),f(y))o_A(x, y) \leq o_B(f(x), f(y))

for all xAx \in A and yAy \in A.

Examples

  • Let MM be Ω\Omega, the set of truth values, so that AA and BB are preorders. The Ω\Omega-enriched monotones between AA and BB are monotones between AA and BB.

  • Let MM be ( 0,,+,0)(\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 00, so that AA and BB are quasipseudometric spaces. The 0\mathbb{R}_{\geq 0}-enriched monotones between AA and BB are distance-decreasing maps between AA and BB.

See also

Last revised on May 31, 2022 at 08:30:39. See the history of this page for a list of all contributions to it.