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Definition

In set theory

Let $\mathbb{Q}$ be the rational numbers $\mathbb{Q}$ and let

be the positive rational numbers.

Let us define a Cauchy algebra to be a set $A$ with a function $\iota \in A^{C(\mathbb{Q})}$, where $C(\mathbb{Q})$ is the set of Cauchy approximations in $\mathbb{Q}$ and $A^{C(\mathbb{Q})}$ is the set of functions with domain $C(\mathbb{Q})$ and codomain $A$, such that

A Cauchy algebra homomorphism is a function $f:B^A$ between Cauchy algebras $A$ and $B$ such that

The category of Cauchy algebras is the category $CauchyAlg$ whose objects $Ob(CauchyAlg)$ are Cauchy algebras and whose morphisms $Mor(CauchyAlg)$ are Cauchy algebra homomorphisms. The set of Cauchy real numbers, denoted $\mathbb{R}_C$, is defined as the initial object in the category of Cauchy algebras.

In homotopy type theory

Let $\mathbb{Q}$ be the rational numbers and let

be the positive rational numbers.

The Cauchy real numbers $\mathbb{R}_C$ are inductively generated by the following:

• a function $\iota: C(\mathbb{Q}) \to \mathbb{R}_C$, where $C(\mathbb{Q})$ is the type of Cauchy approximations in $\mathbb{Q}$.

• a dependent family of terms

  $$a:C(\mathbb{Q}), b:C(\mathbb{Q}) \vdash eq_{\mathbb{R}_C}(a, b): \left( \prod_{\epsilon:\mathbb{Q}_{+}} \vert a_\epsilon - b_\epsilon \vert \lt 4 \epsilon \right) \to (\iota(a) =_{\mathbb{R}_C} \iota(b))$$

• a family of dependent terms

  $$a:\mathbb{R}_C, b:\mathbb{R}_C \vdash \tau(a, b): isProp(a =_{\mathbb{R}_C} b)$$

Comment

The Cauchy real numbers are sometimes defined using sequences $x:\mathbb{N} \to \mathbb{Q}$, $y:\mathbb{N} \to \mathbb{Q}$ with modulus of Cauchy convergence $M:\mathbb{Q}_{+} \to \mathbb{N}$, $N:\mathbb{Q}_{+} \to \mathbb{N}$ respectively. However, the composition of a sequence and a modulus of Cauchy convergence yields a Cauchy approximation, so one could define Cauchy approximations $a:\mathbb{Q}_{+} \to \mathbb{Q}$ as $a \coloneqq x \circ M$ and $b:\mathbb{Q}_{+} \to \mathbb{Q}$ as $b \coloneqq y \circ N$, with $x_{M(\epsilon)} = a_\epsilon$ and $y_{N(\epsilon)} = b_\epsilon$ following from function evaluation. In fact, Cauchy approximations and sequences with a modulus of Cauchy convergence are inter-definable.

See also

• Cauchy real numbers (disambiguation)
• generalized Cauchy real numbers
• HoTT book real numbers
• quotient set
• infinite decimal representation of the Cauchy real numbers?

References

• Auke B. Booij, Analysis in univalent type theory (pdf)
• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)