Homotopy Type Theory Cauchy approximation > history (Rev #4)

Definition

Let $R$ be a Archimedean ordered integral domain with a dense strict order, and let $R_{+}$ be the semiring? of positive terms in $R$.

Suppose $S$ is a $R_{+}$-premetric space, and let $x:R_{+} \to S$ be a net with index type $R_{+}$. 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 linear order, and thus a meet-pseudosemilattice, with $N:R_{+}$ defined as $N \coloneqq \min(\delta, \eta)$ for $\delta:R_{+}$, $\eta:R_{+}$, and meet operation $\min:R_{+} \times R_{+} \to R_{+}$. $\epsilon:R_{+}$ is defined as $\epsilon \coloneqq \delta + \eta$.

Thus, there is a family of dependent terms

A $R_{+}$-Cauchy approximation is the composition $M \circ x$ of a net $x$ and a $R_{+}$-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)