James Hefford

Matt Wilson, James Hefford, Timothée Hoffreumon

Categorical supermaps generalise higher-order quantum operations from finite-dimensional quantum theory to arbitrary circuit theories. In this paper, we establish the Yoneda lemma for categorical supermaps, which states that whenever a physical theory has a suitable notion of channel-state duality, then categorical supermaps on that theory can be concretely represented in terms of that duality. This lemma eliminates any guesswork or ambiguity when defining the appropriate notion of supermap for these theories. As a concrete application, we show that the recently proposed higher-order processes on boxworld can be obtained as a particular instance of categorical supermaps, and put forward a stable definition of higher-order real quantum theory.

Matthew Wilson, James Hefford, Guillaume Boisseau et al.

We build upon our recently introduced concept of an update structure to show that it is a generalisation of very-well-behaved lenses, that is, there is a bijection between a strict subset of update structures and vwb lenses in cartesian categories. We show that update structures are also sufficiently general to capture quantum observables, pinpointing the additional assumptions required to make the two coincide. In doing so, we shift the focus from special commutative dagger-Frobenius algebras to interacting (co)magma (co)module pairs, showing that the algebraic properties of the (co)multiplication arise from the module-comodule interaction, rather than direct assumptions about the magma-comagma pair. We then begin to investigate the zoo of possible update structures, introducing the notions of classical security-flagged databases, and databases of quantum systems. This work is of foundational interest as update structures place previously distinct areas of research in a general class of operationally motivated structures, we expect the taming of this class to illuminate novel relationships between separately studied topics in computer science, physics and mathematics.

James Hefford, Vincent Wang, Matthew Wilson

Modelling concept representation is a foundational problem in the study of cognition and linguistics. This work builds on the confluence of conceptual tools from Gärdenfors semantic spaces, categorical compositional linguistics, and applied category theory to present a domain-independent and categorical formalism of 'concept'.