Dagger categories and the complex numbers: Axioms for the category of finite-dimensional Hilbert spaces and linear contractions

TL;DR

The paper develops an axiomatic, dagger‑rig framework to characterize the category $ extbf{FCon}$ of finite‑dimensional Hilbert spaces and linear contractions, avoiding analytic notions and Solèr’s theorem. By introducing bounded sequential (co)limits as a completeness axiom and a dagger‑finiteness axiom, it separates finite dimensionality from the broader Con setting. It constructs a scalar localisation and a partially ordered semifield of positives, proving these encode the real/complex scalar structure, and proves finite dimensionality and inner products via dagger finiteness. The main result shows $ extbf{D}$ is equivalent to $ extbf{FCon}$, and provides a Solèr‑free route to recover the scalar field as $ ext{R}$ or $ ext{C}$, with potential extensions to broader quaternionic settings. This advances a category‑theoretic foundation for finite‑dimensional quantum structures with minimal analytic assumptions.

Abstract

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.

Theorems & Definitions (97)

• Theorem 1
• Lemma 2
• proof
• Definition 3
• Proposition 4
• proof
• Proposition 5
• Lemma 6
• proof
• Lemma 7