Homotopy Type Theory Cauchy net > history (Rev #8)

Definition
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Definition

In premetric spaces

Let $T$ be a directed type, and let $S$ be a $T$ -premetric space. Given a directed type $I$ , a net $x: I \to S$ is a Cauchy net if

x:I \to S \vdash c(x):\prod_{\epsilon:T} \Vert \sum_{N:I} \prod_{i:I} \prod_{j:I} (i \geq N) \times (j \geq N) \times (x_i \sim_{\epsilon} x_j) \Vert

Cauchy approximations

Let $A$ be a dense Archimedean ordered abelian group with a point $1:A$ and a term $\zeta: 0 \lt 1$ . Let $A_{+} \coloneqq \sum_{a:A} (0 \lt a)$ be the positive cone? of $A$ .

A net $x: A_{+} \to S$ is a $A_{+}$ -Cauchy approximation if

x:R_{+} \to S \vdash c(x):\prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta + \eta} x_\eta

Every $A_{+}$ -Cauchy approximation is a Cauchy net indexed by $A_{+}$ . This is because $A_{+}$ is a strictly ordered type, and thus a directed type and a strictly codirected type, with $N:A_{+}$ defined as $N \coloneqq \delta \otimes \eta$ for $\delta:A_{+}$ and $\eta:A_{+}$ . $\epsilon:R_{+}$ is defined as $\epsilon + \delta + \eta$ .

In Cauchy spaces

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Cauchy sequences

A Cauchy sequence is a Cauchy net whose index type is the natural numbers $\mathbb{N}$ .