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Categories of orthosets and adjointable maps

Jan Paseka, Thomas Vetterlein

TL;DR

The paper develops a categorical framework for orthosets with 0 by focusing on adjointable maps, which generalize Hilbert-space adjoints via the condition $f(x)\perp y\iff x\perp g(y)$. It shows how adjointable maps induce structure-preserving maps on irredundant quotients and on the associated ortholattices, linking orthosets to Dacey spaces and complete orthomodular lattices. It defines two categories, $\mathcal{OS}$ and $\mathcal{iOS}$, analyzes their morphisms (notably the lack of general equalizers and the dagger structure on $\mathcal{iOS}$), and establishes dagger-faithful connections to ortholattice categories. The work bridges projective Hilbert-space geometry with closure-space and lattice-theoretic formalisms, suggesting pathways to describe Hermitian spaces within this orthogonality-centric framework.

Abstract

An orthoset is a non-empty set together with a symmetric and irreflexive binary relation $\perp$, called the orthogonality relation. An orthoset with 0 is an orthoset augmented with an additional element 0, called falsity, which is orthogonal to every element. The collection of subspaces of a Hilbert space that are spanned by a single vector provides a motivating example. We say that a map $f \colon X \to Y$ between orthosets with 0 possesses the adjoint $g \colon Y \to X$ if, for any $x \in X$ and $y \in Y$, $f(x) \perp y$ if and only if $x \perp g(y)$. We call $f$ in this case adjointable. For instance, any bounded linear map between Hilbert spaces induces a map with this property. We discuss in this paper adjointability from several perspectives and we put a particular focus on maps preserving the orthogonality relation. We moreover investigate the category OS of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory iOS of irredundant orthosets with 0. iOS can be made into a dagger category, the dagger of a morphism being its unique adjoint. iOS contains dagger subcategories of various sorts and provides in particular a framework for the investigation of projective Hilbert spaces.

Categories of orthosets and adjointable maps

TL;DR

The paper develops a categorical framework for orthosets with 0 by focusing on adjointable maps, which generalize Hilbert-space adjoints via the condition . It shows how adjointable maps induce structure-preserving maps on irredundant quotients and on the associated ortholattices, linking orthosets to Dacey spaces and complete orthomodular lattices. It defines two categories, and , analyzes their morphisms (notably the lack of general equalizers and the dagger structure on ), and establishes dagger-faithful connections to ortholattice categories. The work bridges projective Hilbert-space geometry with closure-space and lattice-theoretic formalisms, suggesting pathways to describe Hermitian spaces within this orthogonality-centric framework.

Abstract

An orthoset is a non-empty set together with a symmetric and irreflexive binary relation , called the orthogonality relation. An orthoset with 0 is an orthoset augmented with an additional element 0, called falsity, which is orthogonal to every element. The collection of subspaces of a Hilbert space that are spanned by a single vector provides a motivating example. We say that a map between orthosets with 0 possesses the adjoint if, for any and , if and only if . We call in this case adjointable. For instance, any bounded linear map between Hilbert spaces induces a map with this property. We discuss in this paper adjointability from several perspectives and we put a particular focus on maps preserving the orthogonality relation. We moreover investigate the category OS of all orthosets with 0 and adjointable maps between them. We especially focus on the full subcategory iOS of irredundant orthosets with 0. iOS can be made into a dagger category, the dagger of a morphism being its unique adjoint. iOS contains dagger subcategories of various sorts and provides in particular a framework for the investigation of projective Hilbert spaces.
Paper Structure (6 sections, 50 theorems, 21 equations, 1 figure)

This paper contains 6 sections, 50 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Orthosets with $\mathbf 0$
  3. Adjointable maps
  4. Orthometric correspondences
  5. A category of orthosets
  6. A dagger category of irredundant orthosets

Key Result

Lemma 2.3

Let $Y$ be a suborthoset of the orthoset $X$. Assume that, for any $x \in X^{\raisebox{0.2ex}{$\bullet$}}$, there is a subset $A \subseteq Y$ such that $\{x\}^\perp = A^\perp$. Then the maps are mutually inverse isomorphisms.

Figures (1)

  • Figure 1: Example of a complete atomistic orthomodular lattice that does not have the covering property.

Theorems & Definitions (111)

  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 101 more