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

Showing changes from revision #7 to #8: Added | ~~Removed~~ | ~~Chan~~ged

Definition
See also

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 ~~abelian~~ dense ~~group~~ ~~with~~ a ~~point~~ ~~$1:A$~~ Archimedean ordered abelian group , with a point~~strict order~~ $1:A$ and a term $ζ : 0 < 1$ \zeta: 0 \lt 1 , . a Let ~~term~~ $ζ A_{+} : ≔ \sum_{a : A} (0 < 1 a)$ ~~\zeta:~~ A_{+} 0 \coloneqq \sum_{a:A} (0 \lt 1 a) ~~and~~ be a the ~~family~~ of ~~dependent~~ ~~terms~~positive cone? of $A$ .

~~$a:A, b:A \vdash \alpha(a, b):(0 \lt a) \times (0 \lt b) \to (0 \lt a + b)$~~

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

~~Let $A_{+} \coloneqq \sum_{a:A} (0 \lt a)$ be the positive cone? of $A$ .~~

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

A Every ~~net~~ $x : A_{+} \to S$ x: A_{+} ~~\to~~ S -Cauchy approximation is a ~~$A_{+}$ -Cauchy approximation~~Cauchy net if 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 $\epsilon + \delta + \eta$ .

~~$x:R_{+} \to S \vdash c(x):\prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta + \eta} x_\eta$~~
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 $\epsilon + \delta \oplus \eta$ .

In Cauchy spaces

…

Cauchy sequences

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