Introduction to Quantum Combinatorics
Tomasz Maszczyk
TL;DR
This work develops a categorical framework for quantum combinatorics by embedding the classical topos of sets into the presheaf topos $q\mathcal{S}et$ of counital coalgebras and introducing a partial-monoidal quantum Cartesian product. By interpreting the internal logic inside $q\mathcal{S}et$, it recovers the Birkhoff–von Neumann quantum propositional calculus, extends to quantum Boolean algebras, and defines a probabilistic state formalism for observables, underpinning a bottom-up approach to quantum logic. It then introduces quantum quivers and provides a categorified Cuntz–Pimsner construction, proving that the EM category of the CP monad is equivalent to stable representations of the quiver and yielding a categorified Leavitt path algebra. The framework connects classical combinatorics with quantum coalgebraic structures through adjunction-based categorification, offering new insights into quantum graphs, quantum Propositional calculus, and their operator-algebraic analogues within a purely categorical setting.
Abstract
We construct a topos of quantum sets and embed into it the classical topos of sets. We show that the internal logic of the topos of sets, when interpreted in the topos of quantum sets, provides the Birkhoff-von Neumann quantum propositional calculus of idempotents in a canonical internal commutative algebra of the topos of quantum sets. We extend this construction by allowing the quantum counterpart of Boolean algebras of classical truth values which we introduce and study in detail. We realize expected values of observables in quantum states in our topos of quantum sets as a tautological morphism from the canonical internal commutative algebra to a canonical internal object of affine functions on quantum states. We show also that in our topos of quantum sets one can speak about quantum quivers in the sense of Day-Street and Chikhladze. Finally, we provide a categorical derivation of the Leavitt path algebra of such a quantum quiver and relate it to the category of stable representations of the quiver. It is based on a categorification of the Cuntz-Pimsner algebra in the context of functor adjunctions replacing the customary use of Hilbert modules.