nLab Cauchy approximation

 Idea

A Cauchy approximation is a function which behaves as the composition of a regular Cauchy sequence with its modulus of convergence.

Definition

The original notion of a Cauchy approximation first defined in Booij 2020 uses the rational numbers.

A function $x$ from the positive rational numbers to the real numbers is a Cauchy approximation if for all positive rational numbers $\delta$ and $\epsilon$, the distance between $x_\delta$ and $x_\epsilon$ is less than $\delta + \epsilon$. Explicitly:

$$\forall \delta, \epsilon,\; |x_\delta - x_\epsilon| \lt \delta + \epsilon .$$

In a metric space $S$, a function $x$ from the positive rational numbers to $S$ is a Cauchy approximation under the same condition, now relative to the metric $d$ on that space. Explicitly:

$$\forall \delta, \epsilon,\; d(x_\delta,x_\epsilon) \lt \delta + \epsilon .$$

In a gauge space $S$, a function $x$ from the positive rational numbers to $S$ is a Cauchy approximation if this condition is satisfied for each gauging distance separately. Explicitly:

$$\forall d,\; \forall \delta, \epsilon,\; d(x_\delta,x_\epsilon) \lt \delta + \epsilon .$$

In a rational or real premetric space $S$, a function $x$ from the positive rational numbers to $S$ is a Cauchy approximation if this condition is satisfied for the premetric. Explicitly:

$$\forall \delta, \epsilon,\; x_\delta \sim_{\delta + \epsilon} x_\epsilon .$$

Properties

Every Cauchy approximation is a Cauchy net indexed by the positive rational numbers $\mathbb{Q}_{+}$.

A Cauchy approximation is the composition $x \circ M$ of a net $x$ and a rational modulus of convergence $M$.

Real Cauchy approximations

For metric spaces and gauge spaces, since the modulus of convergence of regular Cauchy sequences usually uses the positive real numbers rather than the positive rational numbers, one can use the positive real numbers instead of the positive rational numbers as the domain of the Cauchy approximation, yielding a notion of a real Cauchy approximation. The original notion of a Cauchy approximation can then be called a rational Cauchy approximation.

A function $x$ from the positive real numbers to the real numbers is a real Cauchy approximation if for all positive rational numbers $\delta$ and $\epsilon$, the distance between $x_\delta$ and $x_\epsilon$ is less than $\delta$ and $\epsilon$. Explicitly:

$$\forall \delta, \epsilon,\; |x_\delta - x_\epsilon| \lt \delta + \epsilon .$$

In a metric space $S$, a function $x$ from the positive real numbers to $S$ is a real Cauchy approximation under the same condition, now relative to the metric $d$ on that space. Explicitly:

$$\forall \delta, \epsilon,\; d(x_\delta,x_\epsilon) \lt \delta + \epsilon .$$

In a gauge space $S$, a function $x$ from the positive real numbers to $S$ is a real Cauchy approximation if this condition is satisfied for each gauging distance separately. Explicitly:

$$\forall d,\; \forall \delta, \epsilon,\; d(x_\delta,x_\epsilon) \lt \delta + \epsilon .$$

In a real premetric space $S$, a function $x$ from the positive real numbers to $S$ is a real Cauchy approximation if this condition is satisfied for the ternary relation. Explicitly:

$$\forall \delta, \epsilon,\; x_\delta \sim_{\delta + \epsilon} x_\epsilon .$$

Limits of Cauchy approximations

Similar to limits of Cauchy sequences, one can define the limit of Cauchy approximations. The definition of a limit are the same for both rational and real Cauchy approximations.

Given a Cauchy approximation $x$, a real number $c$ is a limit of $x$ if for all positive rational numbers $\epsilon$ and $\theta$, the distance between $x_\epsilon$ and $c$ is less than $\epsilon + \theta$. Explicitly:

$$\forall \epsilon, \theta,\; |x_\epsilon - c| \lt \epsilon + \theta .$$

In a metric space $S$, given a Cauchy approximation $x$, an element $c$ in $S$ is a limit of $x$ under the same condition, now relative to the metric $d$ on that space. Explicitly:

$$\forall \epsilon, \theta,\; d(x_\epsilon,c) \lt \epsilon + \theta .$$

In a gauge space $S$, given a Cauchy approximation $x$, an element $c$ in $S$ is a limit of $x$ if this condition is satisfied for each gauging distance separately. Explicitly:

$$\forall d,\; \forall \epsilon, \theta,\; d(x_\epsilon,c) \lt \epsilon + \theta .$$

In a rational or real premetric space $S$, given a Cauchy approximation $x$, an element $c$ in $S$ is a limit of $x$ if this condition is satisfied for the premetric. Explicitly:

$$\forall \epsilon, \theta,\; x_\epsilon \sim_{\epsilon + \theta} c .$$

Related concepts

• Booij premetric space
• Cauchy structure
• modulus of convergence
• Cauchy real numbers
• HoTT book real numbers

References

• Auke Booij, Analysis in Univalent Type Theory (2020) [etheses:10411, pdf, pdf]

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)