Dagger Categories and the Complex Numbers: Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions

We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and linear contractions in terms of simple category-theoretic struct...

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Bibliographic Details
Published inApplied categorical structures Vol. 33; no. 3
Main Authors Di Meglio, Matthew, Heunen, Chris
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Nature B.V 01.06.2025
Subjects
Axioms
Complex numbers
Hilbert space
Real numbers
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ISSN0927-2852
1572-9095
DOI10.1007/s10485-025-09803-5

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Summary:We unravel a deep connection between limits of real numbers and limits in category theory. Using a new variant of the classical characterisation of the real numbers, we characterise the category of finite-dimensional Hilbert spaces and linear contractions in terms of simple category-theoretic structures and properties that do not refer to norms, continuity, or real numbers. This builds on Heunen, Kornell, and Van der Schaaf’s easier characterisation of the category of all Hilbert spaces and linear contractions.
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ISSN:0927-2852
1572-9095
DOI:10.1007/s10485-025-09803-5