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