Quantum Logic in Dagger Kernel Categories

TL;DR

This paper explores quantum logic through categorical logic frameworks, focusing on dagger kernel categories, revealing their broad applicability and logical properties like orthomodularity and adjoint connectives.

Contribution

It introduces a minimal axiomatic approach to quantum logic using dagger kernels, unifying various examples and establishing their categorical and logical properties.

Findings

Categories include relations, partial injections, Hilbert spaces, Boolean algebras

Existence of pullbacks, factorisation, orthomodularity

Derivation of Sasaki hook and and-then connectives as adjoints

Abstract

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, and orthomodularity. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres.