Thomason's Theorem Research Articles

Frame definability in finitely valued modal logics

In this paper we study frame definability in finitely valued modal logics and establish two main results via suitable translations: (1) in finitely valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf. [27, Thm. 8]), and (2) a large family of finitely valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and MV-algebras, or even BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt–Thomason theorem applies immediately to these logics. In particular, we obtain the central result from [26] with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely valued modal logics.

Algebraic cobordism and étale cohomology

Thomason's \'{e}tale descent theorem for Bott periodic algebraic $K$-theory \cite{aktec} is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analog of Thomason's theorem for Weibel's homotopy $K$-theory. This is achieved by amplifying the effects from the case of motivic cohomology, using the slice spectral sequence in the case of the universal example of algebraic cobordism. We also obtain integral versions of these statements: Bousfield localization at \'etale motivic cohomology is the universal way to impose \'etale descent for these theories. As applications, we describe the \'etale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an \'etale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the \'{e}tale versions of these motivic spectra.

Sahlqvist correspondence theory for second-order propositional modal logic

AbstractModal logic with propositional quantifiers (i.e. second-order propositional modal logic ($\textsf {SOPML}$)) has been considered since the early time of modal logic. Its expressive power and complexity are high, and its van Benthem–Rosen theorem and Goldblatt–Thomason theorem have been proved by ten Cate (2006, J. Philos. Logic, 35, 209–223). However, the Sahlqvist theory of $\textsf {SOPML}$ has not been considered in the literature. In the present paper, we fill in this gap. We develop the Sahlqvist correspondence theory for $\textsf {SOPML}$, which covers and properly extends existing Sahlqvist formulas in basic modal logic. We define the class of Sahlqvist formulas for $\textsf {SOMPL}$ step by step in a hierarchical way, each formula of which is shown to have a first-order correspondent over Kripke frames effectively computable by an algorithm $\textsf {ALBA}^{\textsf {SOMPL}}$. In addition, we show that certain $\varPi _2$-rules correspond to $\varPi _2$-Sahlqvist formulas in $\textsf {SOMPL}$, which further correspond to first-order conditions, and that even for very simple $\textsf {SOMPL}$ Sahlqvist formulas, they could already be non-canonical.

Characterising modal definability of team-based logics via the universal modality

We study model and frame definability of various modal logics. Let ML(u⃞+) denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We show that a class of Kripke models is definable in ML(u⃞+) if and only if the class is elementary and closed under disjoint unions and surjective bisimulations. We also characterise the definability of ML(u⃞+) in the spirit of the well-known Goldblatt–Thomason theorem. We show that an elementary class F of Kripke frames is definable in ML(u⃞+) if and only if F is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition we study frame definability relative to finite transitive frames and give an analogous characterisation of ML(u⃞+)-definability relative to finite transitive frames. Finally, we initiate the study of model and frame definability in team-based logics. We study (extended) modal dependence logic, (extended) modal inclusion logic, and modal team logic. We establish strict linear hierarchies with respect to model definability and frame definability, respectively. We show that, with respect to model and frame definability, the before mentioned team-based logics, except modal dependence logic, either coincide with ML(u⃞+) or plain modal logic ML. Thus as a corollary we obtain model theoretic characterisation of model and frame definability for the team-based logics.This article subsumes and extends the conference articles [30] and [31].

First-Order Modal Logic: Frame Definability and a Lindström Theorem

We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic (FML, for short). We first study FML-definable frames and give a version of the Goldblatt–Thomason theorem for this logic. The advantage of this result, compared with the original Goldblatt–Thomason theorem, is that it does not need the condition of ultrafilter reflection and uses only closure under bounded morphic images, generated subframes and disjoint unions. We then investigate Lindstrom type theorems for first-order modal logic. We show that FML has the maximal expressive power among the logics extending FML which satisfy compactness, bisimulation invariance and the Tarski union property.

Local Goldblatt–Thomason theorem

The celebrated theorem proved by Goldblatt and Thomason in 1974 gives necessary and sufficient conditions for an elementary (i.e., first-order definable) class of Kripke frames to be modally definable. Here we obtain a local analogue of this result, where in place of frames we consider n-frames, which are frames with n distinguished worlds. For talking about n-frames, we generalize customary modal formulas to what we call modal expressions. Unlike a modal formula, which is evaluated at a single world of a Kripke model, a modal expression with n individual variables is evaluated at an n-tuple of worlds, just as a first-order formula with n free variables. We introduce operations on n-frames that preserve validity of modal expressions, and show that closure under these operations is a necessary and sufficient condition for an elementary class of n-frames to be modally definable. We also discuss the relationship between modal expressions and hybrid logic and leave open questions.

Highly linked tournaments

A (possibly directed) graph is k-linked if for any two disjoint sets of vertices {x1,…,xk} and {y1,…,yk} there are vertex disjoint paths P1,…,Pk such that Pi goes from xi to yi. A theorem of Bollobás and Thomason says that every 22k-connected (undirected) graph is k-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobás–Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments—there is a function f(k) such that every strongly f(k)-connected tournament is k-linked. The bound on f(k) was reduced to O(klog⁡k) by Kühn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly 452k-connected tournament is k-linked.

Vers une structure de catégorie de modèles à la Thomason sur la catégorie des n-catégories strictes

The purpose of this article is to present ideas towards obtaining a model category structure on the category of small strict n-categories, generalizing the one obtained by Thomason on ordinary categories. Following ideas of Grothendieck and Cisinski, we obtain an “abstract Thomason theorem”, which easily implies the classical Thomason theorem. We deduce a 2-categorical Thomason theorem, an incorrect proof of which has been published by K. Worytkiewicz, K. Hess, P. Parent and A. Tonks. For n>2, we isolate sufficient conditions to obtain an n-categorical Thomason theorem. These conditions will be investigated in further work.

The structure of almost all graphs in a hereditary property

A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n ↦ | P n | , where P n denotes the graphs of order n in P . It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then | P n | = 2 ( 1 − 1 / r + o ( 1 ) ) ( n 2 ) , where r = r ( P ) ∈ N is the so-called ‘colouring number’ of P . However, their results tell us very little about the structure of a typical graph G ∈ P . In this paper we describe the structure of almost every graph in a hereditary property of graphs, P . As a consequence, we derive essentially optimal bounds on the speed of P , improving the Alekseev–Bollobás–Thomason Theorem, and also generalising results of Balogh, Bollobás and Simonovits.

Formal completions and idempotent completions of triangulated categories of singularities

The main goal of this paper is to prove that the idempotent completions of triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic. We also discuss Thomason's theorem on dense subcategories and a relation to the negative K-theory.

An inverse $K$-theory functor

Thomason showed that the K -theory of symmetric monoidal categories models all connective spectra. This paper describes a new construction of a permutative category from a \Gamma -space, which is then used to re-prove Thomason's theorem and a non-completed variant.

Semi-Topological K-Homology and Thomason's Theorem

In this paper, we introduce the “semi-topological K-homology” of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated analytic space Y an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the “Bott inverted” semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of Xan. In combination with a result of Friedlander and the author [12, 3.8], this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason’s celebrated theorem that “Bott inverted” algebraic K-theory with Z/n coefficients coincides with topological K-theory with Z/n coefficients.

Homotopy colimits of diagrams over posets and variations on a theorem of Thomason

We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason's theorem on homotopy colimits over posets. In particular this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen's Theorem A for posets.