Metric monads

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.

Quantitative Algebras and a Classification of Metric Monads

“…We just indicate the (very minor) modifications needed for λ ≥ ℵ 1 in Section 7. Enriched ℵ 1 -accessible monads on Met have also been semantically characterized by Rosický [24] who uses a different concept of a type of algebras. In Remark 4.11 (2) he indicates a relationship to quantitative algebras.…”

“…Thus m is monic and split epic, proving that it is invertible. (4) Every algebra α : 24. The naturality of δ α is obvious.…”

Discrete equational theories

“…Similarly, for ω-ary discrete theory T , ω-Birkhoff subcategories correspond to equational theories consisting of ω-equations. Indeed, in this case, U preserves directed colimits of injections (see [22,Theorem 3.19]).…”

“…Consequently, [22, Corollary 5.2, Corollary 5.3, Lemma 5.4 and Example 5.6] are dubious and probably false. Add that [22,Corrolary 5.3] refers to finitary unconditional equational theories of [18] and [17], i.e., to D ω -theories where D ω consists of finite discrete metric spaces (see 3.6).…”

“…Motivated by by probabilistic programming, Mardare, Panangaden and Plotkin [17,18] introduced quantitative equations and developed universal algebra over metric spaces. In the recent paper [22], we have related their quantitative equational theories to theories of Bourke and Garner [10]. More papers were influenced by [17,18], in particular [1] and [19].…”