Types and forgetfulness in categorical linguistics and quantum mechanics

TL;DR

This paper explores the role of types in categorical models of meaning, revealing that self-similarity in objects is crucial and connecting these structures to Frobenius algebras used in quantum mechanics.

Contribution

It introduces a categorical framework linking typed models of meaning with quantum structures, highlighting the importance of self-similarity and Frobenius algebras.

Findings

Types in categorical models must exhibit self-similarity.

Self-similar structures relate to dagger Frobenius algebras.

Typed connectives give rise to classical structures in quantum models.

Abstract

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 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.