Thea Li | Searcharxiv

arXiv2026

Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach

We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schr\"odinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schr\"odinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces.

math.CT↗

arXiv2024

Fibrational Perspectives on Determinization of Finite-State Automata

Colcombet and Petri\c{s}an argued that automata may be usefully considered from a functorial perspective, introducing a general notion of "V-automaton" based on functors into V. This enables them to recover different standard notions of automata by choosing V appropriately, and they further analyzed the determinization for Rel-automata using the Kleisli adjunction between Set and Rel. In this paper, we revisit Colcombet and Petri\c{s}an's analysis from a fibrational perspective, building on Melli\`es and Zeilberger's recent alternative but related definition of categorical automata as functors satisfying the finitary fiber and unique lifting of factorizations property. In doing so, we improve the understanding of determinization in three regards: Firstly, we carefully describe the universal property of determinization in terms of forward-backward simulations. Secondly, we generalize the determinization procedure for Rel automata using a local adjunction between SpanSet and Rel, which provides us with a canonical forward simulation. Finally, we also propose an alternative determinization based on the multiset relative adjunction which retains paths, and we leverage this to provide a canonical forward-backward simulation.

math.CT↗