一句话：证明可用 c-基本定量方程刻画的定量代数簇恰好是对应基本单子的单子范畴。
论文研究定量 Σ-代数：在带可数元数的签名 Σ 上，代数额外配备度量，且所有操作均为非扩张映射。Mardare、Panangaden 与 Plotkin 曾为此引入正则基数 c 的 c-基本定量方程。本文关注可由 c=ℵ1 的此类方程呈现的定量代数范畴，称为 ω1-varieties，并证明它们恰好是度量空间范畴上由可数基本单子诱导的单子范畴。对有限元签名，对应 c=ℵ0 的 ω-varieties；若将空间限制为超度量空间 UMet，则 ω-varieties 恰好是有限基本单子诱导的单子范畴。结果通过范畴论方法建立簇与单子之间的精确对应。
Quantitative Σ-algebras, where Σ is a signature with countable arities, are Σ-algebras equipped with a metric making all operations nonexpanding. They have been studied by Mardare, Panangaden and Plotkin who also introduced c-basic quantitative equations for regular cardinals c. Categories of quantitative algebras that can be presented by such equations for c = ℵ1 are called ω1-varieties. We prove that they are precisely the monadic categories , where is a countably basic monad on the category of metric spaces For Σ finitary one speaks about ω-varieties for c = ℵ0. If all spaces used are restricted to UMet, the category of ultrametric spaces, then ω-varieties are precisely the monadic categories , where is a finitely basic monad.