Types and forgetfulness in categorical linguistics and quantum mechanics
This work addresses theoretical foundations for modeling meaning in linguistics, with potential cross-disciplinary implications for quantum mechanics, but it is incremental in nature.
The paper investigates the role of types in categorical models of meaning, showing that typed models can compare sentences regardless of grammatical structure, and finds that objects associated with types must exhibit self-similarity, leading to connections with dagger Frobenius algebras used in quantum mechanics.
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.