Dual Adjunction Research Articles

McKinsey-Tarski algebras: An alternative pointfree approach to topology

McKinsey and Tarski initiated the study of interior algebras. We propose complete interior algebras as an alternative pointfree approach to topology. We term these algebras McKinsey-Tarski algebras or simply MT-algebras. Associating with each MT-algebra the lattice of its open elements defines a functor from the category of MT-algebras to the category of frames, which we study in depth. We also study the dual adjunction between the categories of MT-algebras and topological spaces, and show that MT-algebras provide a faithful generalization of topological spaces. Our main emphasis is on developing a unified approach to separation axioms in the language of MT-algebras, which generalizes separation axioms for both topological spaces and frames.

Limits, colimits, and spectra of modelled spaces

It is well-known that the construction of Zariski spectra of (commutative) rings yields a dual adjunction between the category of rings and the category of locally ringed spaces. There are many constructions of spectra of algebras in various contexts giving such adjunctions. Michel Coste unified them in the language of categorical logic by showing that, for an appropriate triple (T0,T,Λ) (which we call a spatial Coste context), each T0-model can be associated with a T-modelled space and that this yields a dual adjunction between the category of T0-models and the category of T-modelled spaces and “admissible” morphisms. However, most of his proofs remain unpublished. In this paper, we introduce an alternative construction of spectra of T0-models, which is reminiscent of Zariski spectra, and give a new proof of Coste adjunction. Moreover, we also extend spectra of T0-models to relative spectra of T0-modelled spaces and prove the existence of limits and colimits in the involved categories of modelled spaces. We can deduce, for instance, that the category of ringed spaces whose stalks are fields is complete and cocomplete.

A pointfree theory of Pervin spaces

We lay down the foundations for a pointfree theory of Pervin spaces. A Pervin space is a set equipped with a bounded sublattice of its powerset, and it is known that these objects characterize those quasi-uniform spaces that are transitive and totally bounded. The pointfree notion of a Pervin space, which we call Frith frame, consists of a frame equipped with a generating bounded sublattice. In this paper we introduce and study the category of Frith frames and show that the classical dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, we do not have an equivalence between the categories of transitive and totally bounded quasi-uniform frames and of Frith frames, but we show that the latter is a full coreflective subcategory of the former. We also explore the notion of completeness of Frith frames inherited from quasi-uniform frames, providing a characterization of those Frith frames that are complete and a description of the completion of an arbitrary Frith frame.

Stone dualities from opfibrations

Stone dualities are dual equivalences between certain categories of algebras and those of topological spaces. A Stone duality is often derived from a dual adjunction between such categories by cutting down unnecessary objects. This dual adjunction is called the fundamental adjunction of the duality, but building it often requires concrete topological arguments. The aim of this paper is to construct fundamental adjunctions generically using (op)fibered category theory. This paper defines an abstract notion of formal spaces (including ordinary topological spaces as the leading example), and gives a construction of a fundamental adjunction between the category of algebras and the category of corresponding formal spaces. Moreover, prove an Adjoint Lifting Theorem in the setting of near 2-fibrations, and construct the fundamental adjunction of Priestley duality.

MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

AbstractIt is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

The Vietoris functor and modal operators on rings of continuous functions

We introduce an endofunctor H on the category baℓ of bounded archimedean ℓ-algebras and show that there is a dual adjunction between the category Alg(H) of algebras for H and the category Coalg(V) of coalgebras for the Vietoris endofunctor V on the category of compact Hausdorff spaces. We prove that Gelfand duality lifts to a dual equivalence between Coalg(V) and the full reflective subcategory Algu(H) of Alg(H). We provide an alternate view of Algu(H) by introducing an endofunctor Hu on the full reflective subcategory of baℓ consisting of uniformly complete objects of baℓ and showing that Alg(Hu) is isomorphic to Algu(H). On the one hand, these results generalize those of [1,19] for the category of coalgebras of the Vietoris endofunctor on the category of Stone spaces. On the other hand, they provide an alternate, more categorical proof of a recent result of [6].

Raney Algebras and Duality for $$T_0$$-Spaces

In this note we adapt the treatment of topological spaces via Kuratowski closure and interior operators on powersets to the setting of $$T_0$$ -spaces. A Raney lattice is a complete completely distributive lattice that is generated by its completely join prime elements. A Raney algebra is a Raney lattice with an interior operator whose fixpoints completely generate the lattice. It is shown that there is a dual adjunction between the category of topological spaces and the category of Raney algebras that restricts to a dual equivalence between $$T_0$$ -spaces and Raney algebras. The underlying idea is to take the lattice of upsets of the specialization order with the restriction of the interior operator of a space as the Raney algebra associated to a topological space. Further properties of topological spaces are explored in the dual setting of Raney algebras. Spaces that are $$T_1$$ correspond to Raney algebras whose underlying lattices are Boolean, and Alexandroff $$T_0$$ -spaces correspond to Raney algebras whose interior operator is the identity. Algebraic description of sober spaces results in algebraic considerations that lead to a generalization of sober that lies strictly between $$T_0$$ and sober.

General affine adjunctions, Nullstellensätze, and dualities

We introduce and investigate a category-theoretic abstraction of the standard “system-solution” adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert's Nullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.

Topological duality via maximal spectrum functor

The Isbell duality tells us a dual equivalence between spatial frames (aka. locales) and sober spaces; it is induced by the prime spectrum functor on frames. In the present paper, we give another dual equivalence induced by the maximal spectrum functor. The Isbell duality subsumes all sober spaces, but not all spaces; the duality shown in this paper subsumes all spaces, but not all sober spaces. Non-sober spaces are particularly important in classical algebraic geometry; they include, inter alia, algebraic varieties in the traditional sense, the points of which can be recovered from their open set frames via the maximal spectrum functor (and cannot via the prime spectrum functor). The duality in this paper is particularly useful for those spaces in algebraic geometry. In addition to the duality induced by maximal spectra, we give a dual adjunction lurking behind it, and an algebraic characterization of having enough points in terms of maximal spectra.Communicated by Jason P. Bell

MV-algebras, infinite dimensional polyhedra, and natural dualities

We connect the dual adjunction between MV-algebras and Tychonoff spaces with the general theory of natural dualities, and provide a number of applications. In doing so, we simplify the aforementioned construction by observing that there is no need of using presentations of MV-algebras in order to obtain the adjunction. We also provide a description of the dual maps that is intrinsically geometric, and thus avoids the syntactic notion of definable map. Finally, we apply these results to better explain the relation between semisimple tensor products and coproducts of MV-algebras, and we extend beyond the finitely generated case the characterisations of strongly semisimple and polyhedral MV-algebras.

Generalizations of the Sweedler Dual

As left adjoint to the dual algebra functor, Sweedler's finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler's construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring $R$ the left adjoint of the dual algebra functor on the category of $R$-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf $R$-algebras, provided that $R$ is noetherian and absolutely flat.

Duality Theory and Categorical Universal Logic: With Emphasis on Quantum Structures

Categorical Universal Logic is a theory of monad-relativised hyperdoctrines (or fibred universal algebras), which in particular encompasses categorical forms of both first-order and higher-order quantum logics as well as classical, intuitionistic, and diverse substructural logics. Here we show there are those dual adjunctions that have inherent hyperdoctrine structures in their predicate functor parts. We systematically investigate into the categorical logics of dual adjunctions by utilising Johnstone-Dimov-Tholen's duality-theoretic framework. Our set-theoretical duality-based hyperdoctrines for quantum logic have both universal and existential quantifiers (and higher-order structures), giving rise to a universe of Takeuti-Ozawa's quantum sets via the tripos-to-topos construction by Hyland-Johnstone-Pitts. The set-theoretical hyperdoctrinal models of quantum logic, as well as all quantum hyperdoctrines with cartesian base categories, turn out to give sound and complete semantics for Faggian-Sambin's first-order quantum sequent calculus over cartesian type theory; in addition, quantum hyperdoctrines with monoidal base categories are sound and complete for the calculus over linear type theory. We finally consider how to reconcile Birkhoff-von Neumann's quantum logic and Abramsky-Coecke's categorical quantum mechanics (which is modernised quantum logic as an antithesis to the traditional one) via categorical universal logic.

Mathematical Morphology on Hypergraphs Using Vertex-Hyperedge Correspondence

The focus of this paper is to develop computationally efficient mathematical morphology operators on hypergraphs. To this aim we consider lattice structures on hypergraphs on which we build morphological operators. We develop a pair of dual adjunctions between the vertex set and the hyperedge set of a hypergraph H, by defining a vertex-hyperedge correspondence. This allows us to recover the classical notion of a dilation/erosion of a subset of vertices and to extend it to subhypergraphs of H. This paper also studies the concept of morphological adjunction on hypergraphs for which both the input and the output are hypergraphs.

A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY

AbstractThe aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $.

Morphological filtering on graphs

We study some basic morphological operators acting on the lattice of all subgraphs of an arbitrary (unweighted) graph G. To this end, we consider two dual adjunctions between the edge set and the vertex set of G. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of G and (ii) to extend it to subgraphs of G. Afterward, we propose several new openings, closings, granulometries and alternate filters acting (i) on the subsets of the edge and vertex set of G and (ii) on the subgraphs of G. The proposed framework is then extended to functions that weight the vertices and edges of a graph. We illustrate with applications to binary and grayscale image denoising, for which, on the provided images, the proposed approach outperforms the usual one based on structuring elements.

The Dual Adjunction between MV-algebras and Tychonoff Spaces

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.

Enriched Logical Connections

In the setting of enriched category theory, we describe dual adjunctions of the form \(L\dashv R:{\mathsf{Spa}}^{op} \longrightarrow{\mathsf{Alg}}\) between the dual of the category Spa of “spaces” and the category Alg of “algebras” that arise from a schizophrenic object Ω, which is both an “algebra” and a “space”. We call such adjunctions logical connections. We prove that the exact nature of Ω is that of a module that allows to lift optimally the structure of a “space” and an “algebra” to certain diagrams. Our approach allows to give a unified framework known from logical connections over the category of sets and analyzed, e.g., by Hans Porst and Walter Tholen, with future applications of logical connections in coalgebraic logic and elsewhere, where typically, both the category of “spaces” and the category of “algebras” consist of “structured presheaves”.

Natural Dualities in Partnership

Traditionally in natural duality theory the algebras carry no topology and the objects on the dual side are structured Boolean spaces. Given a duality, one may ask when the topology can be swapped to the other side to yield a partner duality (or, better, a dual equivalence) between a category of topological algebras and a category of structures. A prototype for this procedure is provided by the passage from Priestley duality for bounded distributive lattices to Banaschewski duality for ordered sets. Moreover, the partnership between these two dualities yields as a spin-off a factorisation of the functor sending a bounded distributive lattice to its natural extension, alias, in this case, the canonical extension or profinite completion. The main theorem of this paper validates topology swapping as a uniform way to create new dual adjunctions and dual equivalences: we prove that, for every finite algebra of finite type, each dualising alter ego gives rise to a partner duality. We illustrate the theorem via a variety of natural dualities, some classic and some less familiar. For lattice-based algebras this leads immediately, as in the Priestley–Banaschewski example, to a concrete description of canonical extensions.

Fundamental results for pointfree convex geometry

Inspired by locale theory, we propose “pointfree convex geometry”. We introduce the notion of convexity algebra as a pointfree convexity space. There are two notions of a point for convexity algebra: one is a chain-prime meet-complete filter and the other is a maximal meet-complete filter. In this paper we show the following: (1) the former notion of a point induces a dual equivalence between the category of “spatial” convexity algebras and the category of “sober” convexity spaces as well as a dual adjunction between the category of convexity algebras and the category of convexity spaces; (2) the latter notion of point induces a dual equivalence between the category of “m-spatial” convexity algebras and the category of “m-sober” convexity spaces. We finally argue that the former notion of a point is more useful than the latter one from a category theoretic point of view and that the former notion of a point actually represents a polytope (or generic point) and the latter notion of a point properly represents a point. We also remark on the close relationships between pointfree convex geometry and domain theory.

Exemplaric Expressivity of Modal Logics

This article investigates expressivity of modal logics for transition systems, multitransition systems, Markov chains and Markov processes, as coalgebras of the powerset, finitely supported multiset, finitely supported distribution and measure functor, respectively. Expressivity means that logically indistinguishable states, satisfying the same formulas, are behaviourally indistinguishable. The investigation is based on the framework of dual adjunctions between spaces and logics and focuses on a crucial injectivity property. The approach is generic both in the choice of systems and modalities, and in the choice of a ‘base logic’. Most of these expressivity results are already known, but the applicability of the uniform setting of dual adjunctions to these particular examples is what constitutes the contribution of the article.