## Definition ## Let $R$ be a dense integral subdomain of the [[rational numbers]] $\mathbb{Q}$ and let $R_{+}$ be the positive terms of $R$. Let $S$ be a $R_{+}$-[[premetric space]]. We define the predicate $$isCauchyApproximation(x) \coloneqq \prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta + \eta} x_\eta$$ $x$ is a __$R_{+}$-Cauchy approximation__ if $$x:R_{+} \to S \vdash c(x): isCauchyApproximation(x)$$ The type of $R_{+}$-Cauchy approximations in $S$ is defined as $$C(S, R_{+}) \coloneqq \sum_{x:R_{+} \to S} isCauchyApproximation(x)$$ ## Properties ## Every $R_{+}$-Cauchy approximation is a [[Cauchy net]] indexed by $R_{+}$. This is because $R_{+}$ is a strictly ordered type, and thus a directed type and a strictly codirected type, with $N:R_{+}$ defined as $N \coloneqq \delta \otimes \eta$ for $\delta:R_{+}$ and $\eta:R_{+}$. $\epsilon:R_{+}$ is defined as $\epsilon \coloneqq \delta + \eta$. Thus, there is a family of dependent terms $$x:R_{+} \to S \vdash n(x): isCauchyApproximation(x) \to isCauchyNet(x)$$ An $R_{+}$-Cauchy approximation is the composition $x \circ M$ of a net $x$ and an $R_{+}$-[[modulus of Cauchy convergence]] $M$. ## See also ## * [[premetric space]] * [[Cauchy structure]] * [[modulus of Cauchy convergence]] * [[Cauchy real numbers]] * [[HoTT book real numbers]] ## References ## * Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)) * Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)