Homotopy Type Theory Cauchy approximation > history (Rev #9)

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

$x$ is a $R_{+}$-Cauchy approximation if

The type of $R_{+}$-Cauchy approximations in $S$ is defined as

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 $\epsilon + \delta \oplus \eta$.

Thus, there is a family of dependent terms

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)

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