Authors:Paolo Saracco Pages: 3 - 28 Abstract: Abstract The Structure Theorem for Hopf modules states that if a bialgebra A is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module M is of the form M coA ⊗A, where M coA denotes the space of coinvariant elements in M. Actually, it has been shown that this result characterizes Hopf algebras: A is a Hopf algebra if and only if every Hopf module M can be decomposed in such a way. The main aim of this paper is to extend this characterization to the framework of quasi-bialgebras by introducing the notion of preantipode and by proving a Structure Theorem for quasi-Hopf bimodules. We will also establish the uniqueness of the preantipode and the closure of the family of quasi-bialgebras with preantipode under gauge transformation. Then, we will prove that every Hopf and quasi-Hopf algebra (i.e. a quasi-bialgebra with quasi-antipode) admits a preantipode and we will show how some previous results, as the Structure Theorem for Hopf modules, the Hausser-Nill theorem and the Bulacu-Caenepeel theorem for quasi-Hopf algebras, can be deduced from our Structure Theorem. Furthermore, we will investigate the relationship between the preantipode and the quasi-antipode and we will study a number of cases in which the two notions are equivalent: ordinary bialgebras endowed with trivial reassociator, commutative quasi-bialgebras, finite-dimensional quasi-bialgebras. PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9408-9 Issue No:Vol. 25, No. 1 (2017)

Authors:Mike Behrisch; Sebastian Kerkhoff; Reinhard Pöschel; Friedrich Martin Schneider; Stefan Siegmund Pages: 29 - 57 Abstract: In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems. PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9409-8 Issue No:Vol. 25, No. 1 (2017)

Authors:Nelson Martins-Ferreira; Andrea Montoli Pages: 59 - 75 Abstract: Abstract We study the so-called “Smith is Huq” condition in the context of S-protomodular categories: two S-equivalence relations centralise each other if and only if their associated normal subobjects commute. We prove that this condition is satisfied by every category of monoids with operations equipped with the class S of Schreier split epimorphisms. Some consequences in terms of characterisation of internal structures are explored. PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9411-1 Issue No:Vol. 25, No. 1 (2017)

Authors:Pier Giorgio Basile Pages: 77 - 82 Abstract: Abstract A characterization of effective étale-descent morphisms in the category M-Ord of M-ordered sets, for a given monoid M, is obtained using the corresponding characterization in the category Cat of small categories. PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9412-0 Issue No:Vol. 25, No. 1 (2017)

Authors:Margarida Carvalho; Lurdes Sousa Pages: 83 - 104 Abstract: Abstract In the category T o p 0 of T 0-spaces and continuous maps, embeddings are just those morphisms with respect to which the Sierpiński space is Kan-injective, and the Kan-injective hull of the Sierpiński space is the category of continuous lattices and maps preserving directed suprema and arbitrary infima. In the category L o c of locales and localic maps, we give an analogous characterization of flat embeddings; more generally, we characterize n-flat embeddings, for each cardinal n, as those morphisms with respect to which a certain finite subcategory is Kan-injective. As a consequence, we obtain similar characterizations of the n-flat embeddings in the category T o p 0, and we show that several well-known subcategories of L o c and T o p 0 are Kan-injective hulls of finite subcategories. Moreover, we show that there is a subcategory of spatial locales whose Kan-injective hull is the entire category L o c. PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9413-z Issue No:Vol. 25, No. 1 (2017)

Authors:Olivia Caramello; Nicholas Wentzlaff Pages: 105 - 126 Abstract: Abstract We describe a geometric theory classified by Connes-Consani’s epicylic topos and two related theories respectively classified by the cyclic topos and by the topos \([{\mathbb N}^{\ast }, \textbf {Set}]\) . PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9414-y Issue No:Vol. 25, No. 1 (2017)

Authors:Alexandru Chirvasitu Pages: 127 - 146 Abstract: Abstract We show that for any two von Neumann algebras M and N, the space of non-unital normal homomorphisms N→M with finite support, modulo conjugation by unitaries in M, is Dedekind complete with respect to the partial order coming from the addition of homomorphisms with orthogonal ranges; this ties in with work by Brown and Capraro, where the corresponding objects are given Banach space structures under various niceness conditions on M and N. More generally, we associate to M a Grothendieck site-type category, and show that presheaves satisfying a sheaf-like condition on this category give rise to Dedekind complete lattices upon modding out unitary conjugation. Examples include the above-mentioned spaces of morphisms, as well as analogous spaces of completely positive or contractive maps. We also study conditions under which these posets can be endowed with cone structures extending their partial additions. PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9415-x Issue No:Vol. 25, No. 1 (2017)

Authors:Peter Schauenburg Pages: 147 - 154 Abstract: Abstract Let H be a ×-bialgebra in the sense of Takeuchi. We show that if H is ×-Hopf, and if H fulfills the finiteness condition necessary to define its skew dual H ∨, then the coopposite of the latter is ×-Hopf as well. If in addition the coopposite ×-bialgebra of H is ×-Hopf, then the coopposite of the Drinfeld double of H is ×-Hopf, as is the Drinfeld double itself, under an additional finiteness condition. PubDate: 2017-02-01 DOI: 10.1007/s10485-015-9418-7 Issue No:Vol. 25, No. 1 (2017)

Authors:Nelson Martins-Ferreira Abstract: Abstract A detailed description of a normalized internal bicategory in the category of groups is derived from the general description of internal bicategories in weakly Mal’tsev categories endowed with a V-Mal’tsev operation in the sense of Pedicchio. The example of bicategory of paths in a topological abelian group is presented. PubDate: 2017-03-24 DOI: 10.1007/s10485-017-9486-y

Authors:Javier J. Gutiérrez; Constanze Roitzheim Abstract: Abstract Consider a Quillen adjunction of two variables between combinatorial model categories from \(\mathcal {C}\times \mathcal {D}\) to \(\mathcal {E}\) , a set \(\mathcal {S}\) of morphisms in \(\mathcal {C}\) and a set \(\mathcal {K}\) of objects in \(\mathcal {C}\) . We prove that there is a localised model structure \(L_{\mathcal {S}}\mathcal {E}\) on \(\mathcal {E}\) , where the local objects are the \(\mathcal {S}\) -local objects in \(\mathcal {E}\) described via the right adjoint. Dually, we show that there is a colocalised model structure \(C_{\mathcal {K}}\mathcal {E}\) on \(\mathcal {E}\) , where the colocal equivalences are the \(\mathcal {K}\) -colocal equivalences in \(\mathcal {E}\) described via the right adjoint. These localised and colocalised model structures generalise left and right Bousfield localisations of simplicial model categories, Barnes and Roitzheim’s familiar model structures, and Barwick’s enriched left and right Bousfield localisations. PubDate: 2017-03-16 DOI: 10.1007/s10485-017-9485-z

Authors:Jan Foniok; Claude Tardif Abstract: Abstract We survey results on Hedetniemi’s conjecture which are connected to adjoint functors in the “thin” category of graphs, and expose the obstacles to extending these results. PubDate: 2017-03-14 DOI: 10.1007/s10485-017-9484-0

Authors:Edouard Balzin Abstract: Abstract The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal’s Γ-spaces. The formalism of topological operads generalises well to different categories yielding such notions as \(\mathbb E_n\) -algebras in chain complexes, while the Γ-space approach faces difficulties. In this paper we discuss how, by attempting to extend the Segal approach to arbitrary categoires, one arrives to the problem of understanding “weak” sections of a homotopical Grothendieck fibration. We propose a model for such sections, called derived sections, and study the behaviour of homotopical categories of derived sections under the base change functors. The technology developed for the base-change situation is then applied to a specific class of “resolution” base functors, which are inspired by cellular decompositions of classifying spaces. For resolutions, we prove that the inverse image functor on derived sections is homotopically full and faithful. PubDate: 2017-01-28 DOI: 10.1007/s10485-017-9483-1

Authors:Driss Bennis; J. R. García Rozas; Luis Oyonarte Abstract: Abstract In this paper we are interested in studying the stability question of subcategories of an abelian category \(\mathcal {A}\) constituted of all objects that admit (proper) coproper resolutions (resp. (coproper) proper coresolutions) with terms in a subcategory \(\mathcal {E}\) of \(\mathcal {A}\) . Using a new approach, we give an affirmative answer to the stability question on these categories under the condition that \(\mathcal {E}\) is closed under finite direct sums. This result generalizes Huang’s affirmative answer to the well-known stability question of Gorenstein categories raised by Sather-Wagstaff, Sharif and White. We end the paper with an example showing that the condition imposed on \(\mathcal {E}\) cannot be dropped. PubDate: 2017-01-20 DOI: 10.1007/s10485-016-9478-3

Authors:S. Gorchinskiy; V. Guletskiĭ Abstract: We give a general method of constructing positive stable model structures for symmetric spectra over an abstract simplicial symmetric monoidal model category. The method is based on systematic localization, in Hirschhorn’s sense, of a certain positive projective model structure on spectra, where positivity basically means the truncation of the zero level. The localization is by the set of stabilizing morphisms or their truncated version. PubDate: 2017-01-07 DOI: 10.1007/s10485-016-9480-9

Authors:Themba Dube Abstract: Abstract Let L be a completely regular frame, \(\mathfrak {B}L\) be its Booleanization, υ L be its Hewitt realcompactification, and λ L its Lindelöf coreflection. We characterize those L for which \(\mathfrak {B}(\upsilon L)\cong \upsilon (\mathfrak {B}L)\) , and those for which \(\mathfrak {B}(\lambda L)\cong \lambda (\mathfrak {B}L)\) . In the first case they are precisely those in which every prime ideal of the cozero part with a dense join has a countable subset with a dense join. In the latter case, they are exactly those in which every subset of the frame with a dense join has a countable subset with a dense join. PubDate: 2016-12-24 DOI: 10.1007/s10485-016-9479-2

Authors:Mehmet Akif Erdal; Özgün Ünlü Abstract: Abstract In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gröthendieck group. PubDate: 2016-12-24 DOI: 10.1007/s10485-016-9477-4

Authors:Guram Bezhanishvili; Patrick J. Morandi Abstract: Abstract In our previous paper, in order to develop the pointfree theory of compactifications of ordered spaces, we introduced the concept of a proximity on a biframe as a generalization of the concept of a strong inclusion on a biframe. As a natural next step, we introduce the concept of a proximity morphism between proximity biframes. Like in the case of de Vries algebras and proximity frames, we show that the proximity biframes and proximity morphisms between them form a category PrBFrm in which composition is not function composition. We prove that the category KRBFrm of compact regular biframes and biframe homomorphisms is a proper full subcategory of PrBFrm that is equivalent to PrBFrm. We also show that PrBFrm is equivalent to the category PrFrm of proximity frames, and give a simple description of the concept of regularization using the language of proximity biframes. Finally, we describe the dual equivalence of PrBFrm and the category Nach of Nachbin spaces, which provides a direct way to construct compactifications of ordered spaces. PubDate: 2016-12-24 DOI: 10.1007/s10485-016-9476-5

Authors:Kamran Sharifi Abstract: Abstract K-theory for σ-C*-algebras (countable inverse limits of C*-algebras) has been investigated by N. C. Phillips (K-Theory 3, 441–478, 1989). We use his representable K-theory to show that the space of Fredholm modular operators with coefficients in an arbitrary unital σ-C*-algebra A, represents the functor X↦RK0(C(X,A)) from the category of countably compactly generated spaces to the category of abelian groups. PubDate: 2016-12-20 DOI: 10.1007/s10485-016-9474-7

Authors:Andrew Salch Abstract: Abstract Given an adjoint pair of functors F, G, the composite GF naturally gets the structure of a monad. The same monad may arise from many such adjoint pairs of functors, however. Can one describe all of the adjunctions giving rise to a given monad? In this paper we single out a class of adjunctions with especially good properties, and we develop methods for computing all such adjunctions, up to natural equivalence, which give rise to a given monad. To demonstrate these methods, we explicitly compute the finitary homological presentations of the free A-module monad on the category of sets, for A a Dedekind domain. We also prove a criterion, reminiscent of Beck’s monadicity theorem, for when there is essentially (in a precise sense) only a single adjunction that gives rise to a given monad. PubDate: 2016-12-03 DOI: 10.1007/s10485-016-9473-8