<p>As already mentioned by Lawvere in his 1973 paper, the characterisation of Cauchy completeness of metric spaces in terms of representability of adjoint distributors amounts to the idempotent-split property of an ordinary category when the governing symmetric monoidal-closed category is changed from the extended real half-line to the category of sets. In this paper, for any commutative quantale <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation> taking the role of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[0,\infty ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, we extend these two characterisations of Lawvere-style completeness from ordinary to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation>-normed categories, that is, to categories enriched in the category of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation>-normed sets, i.e., of sets equipped with a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation>-valued function. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {V}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">V</mi> </math></EquationSource> </InlineEquation>-normed categories.</p>

Notions of Cauchy Completeness for Normed Categories

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2026年5月4日

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Dirk Hofmann,
Walter Tholen

摘要

As already mentioned by Lawvere in his 1973 paper, the characterisation of Cauchy completeness of metric spaces in terms of representability of adjoint distributors amounts to the idempotent-split property of an ordinary category when the governing symmetric monoidal-closed category is changed from the extended real half-line to the category of sets. In this paper, for any commutative quantale $\mathcal {V}$ $V$ taking the role of $[0,\infty ]$ $[0, \infty]$ , we extend these two characterisations of Lawvere-style completeness from ordinary to $\mathcal {V}$ $V$ -normed categories, that is, to categories enriched in the category of $\mathcal {V}$ $V$ -normed sets, i.e., of sets equipped with a $\mathcal {V}$ $V$ -valued function. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important $\mathcal {V}$ $V$ -normed categories.