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Quantum Physics and LinguisticsA Compositional, Diagrammatic Discourse$
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Chris Heunen, Mehrnoosh Sadrzadeh, and Edward Grefenstette

Print publication date: 2013

Print ISBN-13: 9780199646296

Published to Oxford Scholarship Online: May 2013

DOI: 10.1093/acprof:oso/9780199646296.001.0001

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Types and Forgetfulness in Categorical Linguistics and Quantum Mechanics

Types and Forgetfulness in Categorical Linguistics and Quantum Mechanics

Chapter:
(p.217) CHAPTER 8 Types and Forgetfulness in Categorical Linguistics and Quantum Mechanics
Source:
Quantum Physics and Linguistics
Author(s):

Peter Hines

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199646296.003.0008

In this chapter, the role of types in categorical models of meaning is investigated. A general scheme for how typed models of meaning may be used to compare sentences, regardless of their grammatical structure—based on other chapters in this book—is described, and a toy example is used as an illustration. Taking as a starting point the question of whether the evaluation of such a type system ‘loses information’, we consider the parametrized typing associated with connectives from this viewpoint. The answer to this question implies that, within full categorical models of meaning, the objects associated with types must exhibit a simple but subtle categorical property known as self-similarity. We investigate the category theory behind this, with explicit reference to typed systems, and their monoidal closed structure. We then demonstrate close connections between such self-similar structures and dagger Frobenius algebras. In particular, we demonstrate that the categorical structures implied by the polymorphically typed connectives give rise to a (lax unitless) form of the special forms of Frobenius algebras known as classical structures, used heavily in abstract categorical approaches to quantum mechanics.

Keywords:   category theory, categorical quantum mechanics, Frobenius algebras, polymorphisms, type systems, models of meaning, sentence similarity

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