Recursive Difference Categories and Topos-Theoretic Universality
Ballús Santacana, Andreu (Universitat Autònoma de Barcelona. Departament de Filosofia)

Data:	2025
Descripció:	18 pàg.
Resum:	We introduce a radically minimal categorical foundation for logic, semantics, and computation, built from a single generative axiom of recursive difference. From the null mnema M0 and iterated labeled extensions by D, we form the free category M and its sheaf topos Sh(M). We prove: Modal completeness: Lawvere-Tierney topologies on Sh(M) classify all standard modal logics (K, T, S4, S5) purely via submonoids of the free monoid D*. Fixed-point expressivity: The internal mu-calculus over infinite branching realizes the full Janin-Walukiewicz theorem. ZFC and Set-modeling: Sh(M) embeds Set via constant sheaves and internalizes a model of ZFC by recursive descent. Turing encodability: Finite-automaton and Turing-machine sheaves arise syntactically, yielding a fully mechanizable internal semantics. Internal meta-theorems: Godel completeness and Lowenheim-Skolem hold internally via total descent and vanishing first cohomology H1. We further construct faithful geometric embeddings: Set -> Sh(M) -> Eff, and Sh(M) -> sSet, connecting to realizability and simplicial frameworks. Unlike HoTT and classical site-theoretic models, Sh(M) exhibits total cohomological triviality, no torsors, and fully conservative gluing of all local data. Thus, we realize Lawvere's vision of deriving semantics-modal, set-theoretic, computational, and meta-logical-entirely from one syntactic axiom, unifying logic, semantics, and computation under a single recursive principle.
Drets:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Llengua:	Anglès
Document:	Working paper ; recerca ; Versió de l'autor
Matèria:	Topos theory ; Categorical logic ; Sheaf semantics ; Modal logic ; Lawvere-Tierney topology ; µ-calculus ; Recursion theory ; Symbolic memory

Adreça original: https://arxiv.org/abs/2505.22931v1

Preprint 18 p, 468.7 KB

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