Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2309.10422

Lifting star-autonomous structures

Luigi Santocanale·Cédric de Lacroix·Gregory Chichery
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a functor $Q$ from a category $C$ to the category $Pos$ of ordered sets and order-preserving functions, we study liftings of various kind of structures from the base category $C$ to the total(or Grothendieck) category $\int Q$. That lifting a monoidal structure corresponds to giving some lax natural transformation making $Q$ almost monoidal, might be part of folklore in category theory.We rely on and generalize the tools supporting this correspondence so to provide exact conditions for lifting symmetric monoidal closed and star-autonomous structures.A corollary of these characterizations is that, if $Q$ factors as a monoidal functor through $SLatt$, the category of complete lattices and sup-preserving functions, then $\int Q$ is always symmetric monoidalclosed. In this case, we also provide a method, based on the double negation nucleus from quantale theory, to turn $\int Q$ into a star-autonomous category.The theory developed, originally motivated from the categories $P-Set$ of Schalk and de Paiva, yields a wide generalization of Hyland and Schalk construction of star-autonomous categories by means of orthogonality structures.

Explore related subjects

cs.LOmath.CTmath.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Luigi Santocanale, Cédric de Lacroix, Gregory Chichery. 2023-09-19. Lifting star-autonomous structures. https://arxiv.org/abs/2309.10422

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Completeness of Kozen's Axiomatization for the Modal mu-Calculus: A Simple Proof

The modal mu-calculus, introduced by Dexter Kozen, is an extension of modal logic with fixpoint operators. Its axiomatization, Koz, was introduced at the same time and is an extension of the minimal modal logic K with the so-called Park fixpoint induction principle. It took more than a decade for the completeness of Koz to be proven, finally achieved by Igor Walukiewicz. However, his proof is fairly involved. In this article, we present an improved proof for the completeness of Koz which, although similar to the original, is simpler and easier to understand. Keywords: The modal mu-calculus, completeness, omega-automata.

cs.LO↗
arXiv2026

Blurred Drinker Paradoxes and Blurred Choice Axioms: Constructive Reverse Mathematics of the Downward Löwenheim-Skolem Theorem

In the setting of constructive reverse mathematics, we analyse the downward Löwenheim-Skolem (DLS) theorem of first-order logic, stating that every infinite model has a countable elementary submodel. Refining the well-known equivalence of the DLS theorem to the axiom of dependent choice (DC) over classical base theories, our constructive approach allows for several finer logical decompositions: Just assuming countable choice (CC), the DLS theorem is equivalent to the conjunction of DC with a newly identified fragment of the excluded middle (LEM) that we call the blurred drinker paradox (BDP). Further without CC, the DLS theorem is equivalent to the conjunction of BDP with similarly blurred weakenings of DC and CC. Independently of their connection with the DLS theorem, we also study BDP and the blurred choice axioms on their own, for instance by showing that BDP is LEM without a contribution of Markov's principle and that blurred DC is DC without a contribution of CC. The paper is hyperlinked with an accompanying Coq development.

cs.LO↗
arXiv2026

Algorithmic Unverifiability of Safety for Fixed and Recursively Self-Improving Systems

We establish mathematical limits of algorithmic safety verification for Turing-complete self-modifying systems, the class in which recursive self-improvement takes place, both for a fixed system and across its own modification. Statically, no verifier is sound, complete and tractable: over unbounded domains by Rice's and Gödel's theorems, over all finite configurations by Trakhtenbrot's theorem, and over succinctly described finite environments because verifying a policy against an adversary is coNP-complete and synthesising one is PSPACE-complete. Dynamically, we model one step of self-modification as a computable transformation of code and ask whether a safety property survives it. If the transformation depends only on behaviour, this is Rice's theorem one level up; if it reads the code, as self-modification does, the question is no longer semantic, yet the same s-m-n reduction works inside a class of behaviourally identical programs and inherits the halting degree. One step is never harder than the property; persistence along the whole trajectory can be $Π^0_2$-complete. Certification by a total algorithm is possible only for transformations of restricted expressivity, not merely for systems that stop changing. No tower of supervisors helps, and every total supervisor errs on an undecidable set of systems. For effectively pointwise properties, every faithful bounded scheme that certifies on finite behavioural evidence admits evolution traces certified at every stage while the property is violated. What survives is exact: a monitor that raises an alarm on violation semidecides it, and comparison against a frozen reference keeps the full theory.

cs.LO↗