The Safari of Update Structures: Visiting the Lens and Quantum Enclosures

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.