Homotopy Type Theory
Cauchy approximation > history (Rev #11)
Contents
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Definition
In set theory
A Cauchy approximation in a premetric space S S is a function x ∈ S ℚ + x \in S^{\mathbb{Q}_{+}} , where S ℚ + S^{\mathbb{Q}_{+}} is the set of functions with domain ℚ + \mathbb{Q}_{+} and codomain S S , such that
∀ δ ∈ ℚ + . ∀ η ∈ ℚ + . x ( δ ) ∼ δ + η x ( η ) \forall \delta \in \mathbb{Q}_{+}.\forall \eta \in \mathbb{Q}_{+}.x(\delta) \sim_{\delta + \eta} x(\eta)
The set of all Cauchy approximations is defined as
C ( S ) ≔ { x ∈ S ℚ + | ∀ δ ∈ ℚ + . ∀ η ∈ ℚ + . x ( δ ) ∼ δ + η x ( η ) } C(S) \coloneqq \{x \in S^{\mathbb{Q}_{+}} \vert \forall \delta \in \mathbb{Q}_{+}.\forall \eta \in \mathbb{Q}_{+}.x(\delta) \sim_{\delta + \eta} x(\eta)\}
In homotopy type theory
Let R R be a dense integral subdomain of the rational numbers ℚ \mathbb{Q} and let R + R_{+} be the positive terms of R R .
Let S S be a R + R_{+} -premetric space . We define the predicate
isCauchyApproximation ( x ) ≔ ∏ δ : R + ∏ η : R + x δ ∼ δ + η x η isCauchyApproximation(x) \coloneqq \prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta + \eta} x_\eta
x x is a R + R_{+} -Cauchy approximation if
x : R + → S ⊢ c ( x ) : isCauchyApproximation ( x ) x:R_{+} \to S \vdash c(x): isCauchyApproximation(x)
The type of R + R_{+} -Cauchy approximations in S S is defined as
C ( S , R + ) ≔ ∑ x : R + → S isCauchyApproximation ( x ) C(S, R_{+}) \coloneqq \sum_{x:R_{+} \to S} isCauchyApproximation(x)
Properties
Every R + R_{+} -Cauchy approximation is a Cauchy net indexed by R + R_{+} . This is because R + R_{+} is a strictly ordered type, and thus a directed type and a strictly codirected type, with N : R + N:R_{+} defined as N ≔ δ ⊗ η N \coloneqq \delta \otimes \eta for δ : R + \delta:R_{+} and η : R + \eta:R_{+} . ϵ : R + \epsilon:R_{+} is defined as ϵ ≔ δ + η \epsilon \coloneqq \delta + \eta .
Thus, there is a family of dependent terms
x : R + → S ⊢ n ( x ) : isCauchyApproximation ( x ) → isCauchyNet ( x ) x:R_{+} \to S \vdash n(x): isCauchyApproximation(x) \to isCauchyNet(x)
An R + R_{+} -Cauchy approximation is the composition x ∘ M x \circ M of a net x x and an R + R_{+} -modulus of Cauchy convergence M M .
See also
References
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