The Way of the Infinitesimal
A short tutorial on non-standard analysis, made in particular for people working in the Categorical Quantum Mechanics crowd.
math.LO↗
arXiv subjects
Fabrizio Genovese
Publications and source records attributed to Fabrizio Genovese.
Custom Hypergraph Categories via Generalized Relations
Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical systems and network theory. When investigating a new application, the question arises of how to identify a suitable process theoretic model. We present a conceptually motivated parameterized framework for the construction of models for process theories. Our framework generalizes the notion of binary relation along four axes of variation, the truth values, a choice of algebraic structure, the ambient mathematical universe and the choice of proof relevance or provability. The resulting categories are preorder-enriched and provide analogues of relational converse and taking graphs of maps. Our constructions are functorial in the parameter choices, establishing mathematical connections between different application domains. We illustrate our techniques by constructing many existing models from the literature, and new models that open up ground for further development.
cs.LO↗
Infinite-dimensional Categorical Quantum Mechanics
We use non-standard analysis to define a category $^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We show the existence of unital special commutative $\dagger$-Frobenius algebras, and we conclude $^\star\!\operatorname{Hilb}$ to be compact closed, with partial traces and a Hilbert-Schmidt inner product on morphisms. We exemplify our techniques on the textbook case of 1-dimensional wavefunctions with periodic boundary conditions: we show the momentum and position observables to be well defined, and to give rise to a strongly complementary pair of unital commutative $\dagger$-Frobenius algebras.
quant-ph↗
Interacting Conceptual Spaces
We propose applying the categorical compositional scheme of [6] to conceptual space models of cognition. In order to do this we introduce the category of convex relations as a new setting for categorical compositional semantics, emphasizing the convex structure important to conceptual space applications. We show how conceptual spaces for composite types such as adjectives and verbs can be constructed. We illustrate this new model on detailed examples.
cs.AI↗