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Axioms for the category of sets and relations

Andre Kornell

TL;DR

This work axiomatizes the dagger category of sets and relations Rel by introducing A–G to capture the shared structure of Rel and Hilb and to mirror axioms from dagger Hilbert space frameworks. It develops notions of dagger kernels, dagger biproducts, and complete semirings, showing how endomorphisms form a rich enrichment while scalars collapse to a two-element Boolean algebra under an Eilenberg–Swindle argument. The authors prove a reconstruction-type result: any dagger compact closed category satisfying the stated hypotheses is dagger-equivalent to Rel, and conversely Rel satisfies the axioms, yielding a Lawvere-style bridge between Rel and Hilb. The results illuminate how relational logic and set-based structure can emulate Hilbert-space-like dagger properties, enabling a unified categorical perspective on these paradigms.

Abstract

We provide axioms for the dagger category of sets and relations that recall recent axioms for the dagger category of Hilbert spaces and bounded operators.

Axioms for the category of sets and relations

TL;DR

This work axiomatizes the dagger category of sets and relations Rel by introducing A–G to capture the shared structure of Rel and Hilb and to mirror axioms from dagger Hilbert space frameworks. It develops notions of dagger kernels, dagger biproducts, and complete semirings, showing how endomorphisms form a rich enrichment while scalars collapse to a two-element Boolean algebra under an Eilenberg–Swindle argument. The authors prove a reconstruction-type result: any dagger compact closed category satisfying the stated hypotheses is dagger-equivalent to Rel, and conversely Rel satisfies the axioms, yielding a Lawvere-style bridge between Rel and Hilb. The results illuminate how relational logic and set-based structure can emulate Hilbert-space-like dagger properties, enabling a unified categorical perspective on these paradigms.

Abstract

We provide axioms for the dagger category of sets and relations that recall recent axioms for the dagger category of Hilbert spaces and bounded operators.
Paper Structure (5 sections, 20 theorems, 36 equations)

This paper contains 5 sections, 20 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Dagger symmetric monoidal categories
  3. Infinite biproducts and complete semirings
  4. Complete Boolean algebras
  5. Characterizations of $\mathsf{Rel}$

Key Result

Theorem 1.1

Let $(\mathsf{C}, \otimes, I, \dag)$ be a dagger symmetric monoidal category that satisfies axioms A--G. Then,

Theorems & Definitions (44)

  • Theorem 1.1
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • Definition 4.4
  • Proposition 4.5
  • proof
  • ...and 34 more