Quotient-Comprehension Chains

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.

In Proceedings QPL 2015, arXiv:1511.01181

References in corpus (4)

• Lüders Theorem for Unsharp Quantum Measurements
• From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality
• New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic
• Total and Partial Computation in Categorical Quantum Foundations

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