Metric monads

Jiří Rosický

doi:10.1017/s0960129521000220

Abstract

Abstract We develop universal algebra over an enriched category and relate it to finitary enriched monads over . 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.

Highlights

Classical universal algebra specifies algebras by means of finitary operations and equations
Our aim is to get a better understanding of metric universal algebra, i.e., universal algebra over metric spaces
Metric universal algebra was introduced in Mardare et al (2016) where quantitative equations =ε were used instead of equations

Summary

Introduction

Classical universal algebra specifies algebras by means of finitary operations and equations. Over Met, we can apply Di Liberti and Rosický (2009) and relate equational theories whose operations have finite metric spaces as arities to monads preserving directed colimits of isometries. Both ordered algebras and quantitative algebras form computationally relevant structures making possible an algebraic approach to the semantics of programming languages and computational effects. We show how it can be used to prove the result from Bourke and Garner (2019) that λ-ary pretheories over a locally λpresentable category K correspond to λ-ary monads on K We apply it to the case when K is not locally finitely presentable but only locally finitely generated in the sense of Di Liberti and Rosický (2009), which is the case of metric spaces.

Preliminaries about Categories and Metric Spaces

Algebras in General Categories

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