Applied Categorical Structures (APPL CATEGOR STRUCT)

Publications in this journal

• Article: γ-Frames, GΓ-Algebras and Measurable Spaces

  Helmut Röhrl
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  ABSTRACT: The category of γ -Frm of γ-frames, which is isomorphic to the category GΓ-Alg of \mathbb D\mathbb D-algebras satisfying certain identities, and the category γ -Top of γ-topological spaces provide the background for the category γ -Mbl of γ-measurable spaces. As in the category of frames, the functor Wg: g\text - Top ' (X,WX) ® W Î g\text - Frmop\Omega_\gamma: \gamma {\text{ - }}Top \ni (X,\Omega_X) \mapsto \Omega_\S \in \gamma {\text{ - }}Frm^{op} has a right adjoint Ptg: GG\text - Algop ® g\text - TopPt_\gamma: G\Gamma {\text{ - }}Alg^{op} \to \gamma {\text{ - }}Top.
  Applied Categorical Structures 05/2012; 18(1):31-53.
• Article: Constructing New Braided T-Categories over Weak Hopf Algebras

  Ling Liu, Shuanhong Wang
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  ABSTRACT: Let Aut weak Hopf (H) denote the set of all automorphisms of a weak Hopf algebra H with bijective antipode in the sense of Böhm et al. (J Algebra 221:385–438, 1999) and let G be a certain crossed product group Aut weak Hopf (H)×Aut weak Hopf (H). The main purpose of this paper is to provide further examples of braided T-categories in the sense of Turaev (1994, 2008). For this, we first introduce a class of new categories HWYDH(a, b) _{H}{\mathcal {WYD}}^{H}(\alpha, \beta) of weak (α, β)-Yetter-Drinfeld modules with α, β ∈ Aut weak Hopf (H) and we show that the category WYD(H) = {HWYDH(a, b)}(a, b) Î G{\mathcal WYD}(H) =\{{}_{H}\mathcal {WYD}^{H}(\alpha, \beta)\}_{(\alpha , \beta )\in G} becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic (Isr J Math 158:349–365, 2007). Finally, when H is finite-dimensional we construct a quasitriangular weak T-coalgebra WD(H) = {WD(H)(α, β)}(α, β) ∈ G in the sense of Van Daele and Wang (Comm Algebra, 2008) over a family of weak smash product algebras {[`(H*cop# H(a,b))]}(a, b) Î G\{\overline{H^{*cop}\# H_{(\alpha,\beta)}}\}_{(\alpha , \beta)\in G}, and we obtain that WYD(H){\mathcal {WYD}}(H) is isomorphic to the representation category of the quasitriangular weak T-coalgebra WD(H). KeywordsWeak Hopf algebra-Braided T-category-Weak (α, β)-Yetter-Drinfeld modules -Quasitriangular weak T-coalgebra Mathematics Subject Classifications (2000)16W30
  Applied Categorical Structures 05/2012; 18(4):431-459.
• Article: Lattice-ordered Fields Determined by d-elements

  Jingjing Ma, R. H. Redfield
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  ABSTRACT: Most results on the structure of lattice-ordered fields require that the field have a positive multiplicative identity. We construct a functor from the category of lattice-ordered fields with a vector space basis of d-elements to the full subcategory of such fields with positive multiplicative identities. This functor is a left adjoint to the forgetful functor and, in many cases, allows us to write all compatible lattice orders in terms of orders with positive multiplicative identities. We also use these results to characterize algebraically those extensions of totally ordered fields that have vℓ-bases of d-elements.
  Applied Categorical Structures 05/2012; 15(1):19-33.
• Article: A Note on the Five Lemma

  Friday Ifeanyi Michael
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  ABSTRACT: We formulate and prove a “five lemma”, which unifies two independent generalizations of the classical five lemma in an abelian category: the five lemma in a (modular) semi-exact category in the sense of M. Grandis, and the five lemma in a pointed regular protomodular category in the sense of D. Bourn. KeywordsExact sequence–Five lemma–Cover relation–Homological morphism–Semi-exact category–Regular protomodular category
  Applied Categorical Structures 05/2012;
• Article: Epicompletion in Frames with Skeletal Maps, IV: *-Regular Frames

  Jorge Martínez
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  ABSTRACT: Earlier work has shown that there is a monoreflection ψ of the category of compact normal, joinfit frames with skeletal frame maps in the subcategory consisting of strongly projectable frames. This article extends the domain of ψ to the *-regular frames. The saturation nucleus s is a reflection with respect to weakly closed frame maps, in the subcategory of subfit frames. Moreover, s·ψ = ψ·s, on compact normal, joinfit frames with skeletal, weakly closed frame maps, and s·ψ is an epireflection, but not a monoreflection, in the subcategory of strongly projectable, regular frames, all of which are epicomplete. KeywordsEpicompletion–Skeletal map–Joinfit frame–Monoreflection–Saturation
  Applied Categorical Structures 05/2012; 20(2):189-208.
• Article: Compactification with Respect to a Generalized-net Convergence on Constructs

  Josef Šlapal
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  ABSTRACT: We introduce and deal with a convergence on (objects of) constructs which is expressed in terms of generalized nets. The generalized nets used are obtained from the usual nets by replacing the construct of directed sets and cofinal maps by an arbitrary construct. Convergence separation and convergence compactness are then introduced in a natural way. We study the convergence compactness and compactification and show that they behave in much the same way as the compactness and compactification of topological spaces. KeywordsGeneralized net–Convergence structure on a construct–Categorical closure operator–Separation–Compactness
  Applied Categorical Structures 05/2012; 19(2):523-537.
• Article: A Coherent Homotopy Category of 2-track Commutative Cubes

  K. A. Hardie, K. H. Kamps, P. J. Witbooi
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  ABSTRACT: We consider a category H\ominus Ä{\mathcal H}^{\ominus \otimes} (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of H\ominus Ä{\mathcal H}^{\ominus \otimes} to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation. KeywordsTrack–Semitrack–Homotopy 2-groupoid–Triple category–Homotopy pair–Interchange 2-track–Toda bracket
  Applied Categorical Structures 05/2012; 19(1):39-60.
• Article: Esakia Style Duality for Implicative Semilattices

  Guram Bezhanishvili, Ramon Jansana
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  ABSTRACT: We develop a new duality for implicative semilattices, generalizing Esakia duality for Heyting algebras. Our duality is a restricted version of generalized Priestley duality for distributive semilattices, and provides an improvement of Vrancken-Mawet and Celani dualities. We also show that Heyting algebra homomorphisms can be characterized by means of special partial functions between Esakia spaces. On the one hand, this yields a new duality for Heyting algebras, which is an alternative to Esakia duality. On the other hand, it provides a natural generalization of Köhler’s partial functions between finite posets to the infinite case. KeywordsImplicative semilattice–Heyting algebra–Duality theory
  Applied Categorical Structures 05/2012;
• Article: Convexities Generated by L-Monads

  Taras Radul
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  ABSTRACT: Let \mathbbF\mathbb{F} be a monad in the category Comp. We build for each \mathbbF\mathbb{F}-algebra a convexity in general sense (see van de Vel 1993). We investigate properties of such convexities and apply them to prove that the multiplication map of the order-preserving functional monad is soft. KeywordsMonad–Convexity–Soft map
  Applied Categorical Structures 05/2012; 19(4):729-739.
• Article: The Groupoidal Analogue to Joyal’s Category Θ is a Test Category

  Dimitri Ara
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  ABSTRACT: We introduce the groupoidal analogue [(Q)\tilde]\widetilde{\Theta} to Joyal’s cell category Θ and we prove that [(Q)\tilde]\widetilde{\Theta} is a strict test category in the sense of Grothendieck. This implies that presheaves on [(Q)\tilde]\widetilde{\Theta} model homotopy types in a canonical way. We also prove that the canonical functor from Θ to [(Q)\tilde]\widetilde{\Theta} is aspherical, again in the sense of Grothendieck. This allows us to compare weak equivalences of presheaves on [(Q)\tilde]\widetilde{\Theta} to weak equivalences of presheaves on Θ. Our proofs apply to other categories analogous to Θ. Keywords∞-Category–∞-Groupoid–Cell category–Décalage–Globular extension–Homotopy–Localization–Test category–Weak equivalence
  Applied Categorical Structures 05/2012;
• Article: Normal Subalgebras, I

  Aldo Ursini
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  ABSTRACT: We extend the notions of normal subalgebras, clots and ideals of an algebra A in a variety of (universal) algebras, from the familiar case of a single constant to the case of any number of constants. The first idea is that a subalgebra of A is normal when it is the inverse image under some morphism of the subalgebra generated by constants in the target. We argue that a better approach is obtained by considering pullbacks of γ B and g : A → B, where g : A → B is some morphism and γ B is the morphism from the initial algebra of the variety to B. Examples are shown in Heyting algebras, boolean algebras and unitary rings. Ideals and clots are generalizations of this notion, defined instead by closure under derived operations which have the right behavior on constants. There are several characterizations of these notions; some of them aiming at a categorical generalization. We deal with an (extended) notion of subtractivity, showing that it implies that ideals coincide with normal subalgebras, and it is connected with notions of coherence of congruences, allowing a characterization of protomodular varieties. KeywordsNormal subalgebras–Ideals–Equational classes of algebras
  Applied Categorical Structures 05/2012;
• Article: Enriched Logical Connections

  Alexander Kurz, Jiří Velebil
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  ABSTRACT: In the setting of enriched category theory, we describe dual adjunctions of the form L\dashv R:Spaop ®AlgL\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”. KeywordsLogical connection–Schizophrenic object–Module
  Applied Categorical Structures 05/2012;
• Article: Bifibrations and Weak Factorisation Systems

  Alexandru Emil Stanculescu
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  ABSTRACT: We review a theorem of A. Roig about Quillen model structures on Grothendieck bifibrations and observe that it contains a gap. We reformulate one of its assumptions in order to validate it. As an application to the new version, we introduce the fibred model structure on the category of small categories enriched in a suitable monoidal model category. KeywordsBifibration–Weak factorisation system–Model category–Enriched category
  Applied Categorical Structures 05/2012; 20(1):19-30.
• Article: Another Approach to Connectedness with Respect to a Closure Operator

  J. Šlapal
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  ABSTRACT: We introduce a new concept of connectedness with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject semilattices, naturally generalizes the classical connectedness of topological spaces and we show that it also behaves accordingly. Moreover, as the main result, we prove that the connectedness introduced is preserved, under some natural conditions, by inverse images of subobjects under quotient morphisms. An application of this result in digital topology is discussed too.
  Applied Categorical Structures 05/2012; 17(6):603-612.
• Article: Adjoining an Identity to a Reduced Archimedean f-ring, II: Algebras

  A. W. Hager, D. G. Johnson
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  ABSTRACT: In “Part I” (presented at Ord05 (Oxford, MS)), we have discussed, for reduced archimedean f-rings, the canonical extension of such a ring, A, to one with identity, uA, and the class U of u-extendable maps (i.e., homomorphisms which lift over the u’s to identity preserving homomorphisms). We showed that U is a category and u becomes a functor from U which is a monoreflection; the maps in U were characterized. This paper addresses the interaction between our functor u, and v , the vector lattice monoreflection in archimedean ℓ-groups (due to Conrad and Bleier). In short, v restricts to a monoreflection of reduced archimedean f-rings into reduced archimedean f-algebras, ψ ∈ U if and only if v ψ ∈ U, and vu is a monoreflection into reduced archimedean f-algebras with identity. This work was motivated by the question put to us by G. Buskes at Ord05: what maps are o-extendable; i.e., extend over the orthomorphism rings? (The orthomorphism ring oA is a unital extension of uA, and any o-extendable map lies in U.) While a complete answer seems quite complicated (if not hopelessly out of reach), here we shall identify a class of objects D for which oD = vuD and all maps from D lie in U, hence any map from D to a reduced archimedean f-algebra is o-extendable.
  Applied Categorical Structures 05/2012; 15(1):35-47.
• Article: On the Notion of a Semi-Abelian Category in the Sense of Palamodov

  Yaroslav Kopylov, Sven-Ake Wegner
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  ABSTRACT: In the sense of Palamodov, a preabelian category is semi-abelian if for every morphism the natural morphism between the cokernel of its kernel and the kernel of its cokernel is simultaneously a monomorphism and an epimorphism. In this article we present several conditions which are all equivalent to semi-abelianity. First we consider left and right semi-abelian categories in the sense of Rump and establish characterizations of these notions via six equivalent properties. Then we use these properties to deduce the characterization of semi-abelianity. Finally, we investigate two examples arising in functional analysis which illustrate that the notions of right and left semi-abelian categories are distinct and in particular that such categories occur in nature. KeywordsPreabelian category–Semi-abelian category–Quasi-abelian category–Category of bornological spaces
  Applied Categorical Structures 04/2012;
• Article: Universality of Categories of Coalgebras

  Václav Koubek, Jiří Sichler, Věra Trnková
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  ABSTRACT: Under GCH, a set functor F does not preserve finite unions of non-empty sets if and only if the category Coalg F of all F-coalgebras is universal. Independently of GCH, we show that for any non-accessible functor F preserving intersections, the category Coalg F has a large discrete full subcategory, and we give an example of a category of F-coalgebras that is not universal, yet has a large discrete full subcategory. KeywordsCoalgebra–Full embedding–Universal category
  Applied Categorical Structures 04/2012; 19(6):939-957.
• Article: Notes on δ-Koszul Algebras

  Jia-Feng Lü
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  ABSTRACT: This paper is a continuous work of δ-Koszul algebras, which were first introduced by Green and Marcos in 2005 (see Green and Marcos, Commun Algebra 33(6):1753–1764, 2005). Let Kd(A)\mathcal{K}^{\delta}(A) be the category of δ-Koszul modules. It is proved that Kd(A)\mathcal{K}^{\delta}(A) preserves kernels of epimorphisms if and only if the “minimal Horseshoe Lemma” (“MHL” for short) holds. Further, a special class of δ-Koszul algebras named periodic δ -algebras are introduced, which have close connection with Koszul algebras and provide answers to the questions raised by Green and Marcos (Commun Algebra 33(6):1753–1764, 2005). Finally, we construct new periodic δ-algebras from the given ones in terms of one-point extension and sum-extension. Keywords δ-Koszul algebras– δ-Algebras– δ-Koszul modules–Minimal Horseshoe Lemma
  Applied Categorical Structures 04/2012; 20(2):143-159.
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  Available from: ArXiv

  Article: The Compositional Construction of Markov Processes

  Luisa de Francesco Albasini, Nicoletta Sabadini, Robert F. C. Walters
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  ABSTRACT: We describe a symmetric monoidal category whose arrows are automata in which the actions have probabilities. The endomorphisms of the identity for the tensor are classical finite Markov processes. The operations of the category permit the compositional description Markov processes. We illustrate by describing a Markov process with 12 n states, which represents a model of the classical Dining Philosopher problem with n dining philosophers, showing how to calculate the probability of reaching deadlock in k steps. A straightforward application of the Perron-Frobenius Theorem yields that this probability tends to 1 as k tends to infinity. KeywordsProbabilistic automaton–Symmetric monoidal category–Compact closed–Markov process–Frobenius algebra–Compositionality
  Applied Categorical Structures 04/2012; 19(1):425-437.