Homotopy Type Theory
Cauchy approximation > history (Rev #7)
Definition
Let A A be a abelian group with a point 1 : A 1:A , a strict order < \lt , a term ζ : 0 < 1 \zeta: 0 \lt 1 and a family of dependent terms
a : A , b : A ⊢ α ( a , b ) : ( 0 < a ) × ( 0 < b ) → ( 0 < a + b ) a:A, b:A \vdash \alpha(a, b):(0 \lt a) \times (0 \lt b) \to (0 \lt a + b)
Let A + ≔ ∑ a : A ( 0 < a ) A_{+} \coloneqq \sum_{a:A} (0 \lt a) be the positive cone? of A A .
Let S S be a A + A_{+} -premetric space . We define the predicate
isCauchyApproximation ( x ) ≔ ∏ δ : A + ∏ η : A + x δ ∼ δ ⊕ η x η isCauchyApproximation(x) \coloneqq \prod_{\delta:A_{+}} \prod_{\eta:A_{+}} x_\delta \sim_{\delta \oplus \eta} x_\eta
x x is a A + A_{+} -Cauchy approximation if
x : A + → S ⊢ c ( x ) : isCauchyApproximation ( x ) x:A_{+} \to S \vdash c(x): isCauchyApproximation(x)
The type of A + A_{+} -Cauchy approximations in S S is defined as
C ( S , A + ) ≔ ∑ x : A + → S isCauchyApproximation ( x ) C(S, A_{+}) \coloneqq \sum_{x:A_{+} \to S} isCauchyApproximation(x)
Properties
Every A + A_{+} -Cauchy approximation is a Cauchy net indexed by A + A_{+} . This is because A + A_{+} is a strictly ordered type, and thus a directed type and a strictly codirected type, with N : A + N:A_{+} defined as N ≔ δ ⊗ η N \coloneqq \delta \otimes \eta for δ : A + \delta:A_{+} and η : A + \eta:A_{+} . ϵ : R + \epsilon:R_{+} is defined as ϵ + δ ⊕ η \epsilon + \delta \oplus \eta .
Thus, there is a family of dependent terms
x : A + → S ⊢ n ( x ) : isCauchyApproximation ( x ) → isCauchyNet ( x ) x:A_{+} \to S \vdash n(x): isCauchyApproximation(x) \to isCauchyNet(x)
A A + A_{+} -Cauchy approximation is the composition x ∘ M x \circ M of a net x x and a A + A_{+} -modulus of Cauchy convergence M M .
See also
References
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