Quotient-Comprehension Chains

TL;DR

This paper explores the deep connections between quotients, comprehension, and measurement across various logical frameworks, proposing a new perspective that could unify understanding in categorical logic and quantum theory.

Contribution

It introduces a novel quotient-and-comprehension perspective on measurement instruments and provides diverse examples to guide future unifying theories.

Findings

Quotients and comprehension relate to measurement in quantum, probabilistic, and classical logic.

Examples include both simple and complex cases, such as von Neumann algebras.

The paper lays groundwork for a unifying theory of measurement in categorical logic.

Abstract

Quotients and comprehension are fundamental mathematical constructions that can be described via adjunctions in categorical logic. This paper reveals that quotients and comprehension are related to measurement, not only in quantum logic, but also in probabilistic and classical logic. This relation is presented by a long series of examples, some of them easy, and some also highly non-trivial (esp. for von Neumann algebras). We have not yet identified a unifying theory. Nevertheless, the paper contributes towards such a theory by introducing the new quotient-and-comprehension perspective on measurement instruments, and by describing the examples on which such a theory should be built.