Metric monads

TL;DR

This paper develops a universal algebra framework over enriched categories, relating it to enriched monads, and applies it to ordered and metric universal algebra to enhance understanding of finitary monads in these contexts.

Contribution

It introduces a universal algebra approach over enriched categories and connects it to finitary enriched monads, extending to ordered and metric algebraic structures.

Findings

Unified framework for ordered and metric universal algebra

Relation between universal algebra and enriched monads established

Enhanced understanding of finitary monads on metric spaces

Abstract

We develop universal algebra over an enriched category $\mathcal K$ and relate it to finitary enriched monads over $\mathcal K$. Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.