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 Applied Categorical StructuresJournal Prestige (SJR): 0.49 Number of Followers: 2      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-9095 - ISSN (Online) 0927-2852 Published by Springer-Verlag  [2348 journals]
• Maximal Ideals in Module Categories and Applications
• Authors: Manuel Cortés-Izurdiaga; Alberto Facchini
Pages: 617 - 629
Abstract: We study the existence of maximal ideals in preadditive categories defining an order $$\preceq$$ between objects, in such a way that if there do not exist maximal objects with respect to $$\preceq$$ , then there is no maximal ideal in the category. In our study, it is sometimes sufficient to restrict our attention to suitable subcategories. We give an example of a category $$\mathbf {C}_F$$ of modules over a right noetherian ring R in which there is a unique maximal ideal. The category $$\mathbf {C}_F$$ is related to an indecomposable injective module F, and the objects of $$\mathbf {C}_F$$ are the R-modules of finite F-rank.
PubDate: 2018-08-01
DOI: 10.1007/s10485-017-9505-z
Issue No: Vol. 26, No. 4 (2018)

• Open Maps of Involutive Quantales
• Authors: Pedro Resende
Pages: 631 - 644
Abstract: By a map $$p:Q\rightarrow X$$ of involutive quantales is meant a homomorphism $$p^*:X\rightarrow Q$$ . Calling a map p weakly open if $$p^*$$ has a left adjoint $$p_!$$ which satisfies the Frobenius reciprocity condition (i.e., $$p_!$$ is a homomorphism of X-modules), we say that p is open if it is stably weakly open. We also study a two-sided version, FR2, of the Frobenius reciprocity condition, and show that the weakly open surjections that satisfy FR2 are open. Maps of the latter kind arise in the study of Fell bundles on groupoids.
PubDate: 2018-08-01
DOI: 10.1007/s10485-017-9506-y
Issue No: Vol. 26, No. 4 (2018)

• Strictly Zero-Dimensional Biframes and a Characterisation of Congruence
Frames
• Authors: Graham Manuell
Pages: 645 - 655
Abstract: Strictly zero-dimensional biframes were introduced by Banaschewski and Brümmer as a class of strongly zero-dimensional biframes including the congruence biframes. We consider the category of strictly zero-dimensional biframes and show it is both complete and cocomplete. We characterise the extremal epimorphisms in this category and explore the special position that congruence biframes hold in it. Finally, we provide an internal characterisation of congruence biframes, and hence, of congruence frames.
PubDate: 2018-08-01
DOI: 10.1007/s10485-017-9507-x
Issue No: Vol. 26, No. 4 (2018)

• Ladders of Compactly Generated Triangulated Categories and Preprojective
Algebras
• Authors: Nan Gao; Chrysostomos Psaroudakis
Pages: 657 - 679
Abstract: In this paper we characterize when a recollement of compactly generated triangulated categories admits a ladder of some height going either upwards or downwards. As an application, we show that the derived category of the preprojective algebra of Dynkin type $$\mathbb {A}_n$$ admits a periodic infinite ladder, where the one outer term in the recollement is the derived category of a differential graded algebra.
PubDate: 2018-08-01
DOI: 10.1007/s10485-017-9508-9
Issue No: Vol. 26, No. 4 (2018)

• Rickart and Dual Rickart Objects in Abelian Categories: Transfer via
Functors
• Authors: Septimiu Crivei; Gabriela Olteanu
Pages: 681 - 698
Abstract: We study the transfer of (dual) relative Rickart properties via functors between abelian categories, and we deduce the transfer of (dual) relative Baer property. We also give applications to Grothendieck categories, comodule categories and (graded) module categories, with emphasis on endomorphism rings.
PubDate: 2018-08-01
DOI: 10.1007/s10485-017-9509-8
Issue No: Vol. 26, No. 4 (2018)

• Approximate Injectivity
• Authors: J. Rosický; W. Tholen
Pages: 699 - 716
Abstract: In a locally $$\lambda$$ -presentable category, with $$\lambda$$ a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are $$\lambda$$ -presentable, are known to be characterized by their closure under products, $$\lambda$$ -directed colimits and $$\lambda$$ -pure subobjects. Replacing the strict commutativity of diagrams by “commutativity up to $$\mathcal {\varepsilon }$$ ”, this paper provides an “approximate version” of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed $$\mathcal {\varepsilon }$$ -generalizations of the notion of $$\lambda$$ -purity. The categorical theory is being applied to the locally $$\aleph _1$$ -presentable category of Banach spaces and their linear operators of norm at most 1, culminating in a largely categorical proof for the existence of the so-called Gurarii Banach space.
PubDate: 2018-08-01
DOI: 10.1007/s10485-017-9510-2
Issue No: Vol. 26, No. 4 (2018)

• Grothendieck Categories as a Bilocalization of Linear Sites
• Authors: Julia Ramos González
Pages: 717 - 745
Abstract: Let k be a commutative ring. We prove that the 2-category $$\mathsf {Grt}_k$$ of Grothendieck abelian k-linear categories with colimit preserving k-linear functors and k-linear natural transformations is a bicategory of fractions in the sense of Pronk [17] of the 2-category $$\mathsf {Site}_{k,\mathsf {cont}}$$ of k-linear sites with k-linear continuous functors and k-linear natural transformations. In complete analogy, we prove that the conjugate-opposite 2-category of the 2-category $$\mathsf {Topoi}_k$$ of Grothendieck abelian k-linear categories with k-linear geometric morphisms and k-linear morphisms between them is a bicategory of fractions of the 2-category $$\mathsf {Site}_k$$ of k-linear sites with k-linear morphisms of sites and k-linear natural transformations. In addition, we show how the first statement can potentially be used to make the tensor product of Grothendieck categories from [14] into a bi-monoidal structure on $$\mathsf {Grt}_k$$ .
PubDate: 2018-08-01
DOI: 10.1007/s10485-017-9511-1
Issue No: Vol. 26, No. 4 (2018)

• Extensions of Operators, Liftings of Monads, and Distributive Laws
• Authors: Shilong Zhang; Li Guo; William Keigher
Pages: 747 - 765
Abstract: In a previous study, the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC) is shown to allow extensions of differential and Rota–Baxter operators on the one hand, and to give rise to categorical explanations using the ideas of liftings of monads and comonads, and mixed distributive laws on the other. Generalizing the FFTC, we consider in this paper a class of constraints between a differential operator and a Rota–Baxter operator. For a given constraint, we show that the existences of extensions of differential and Rota–Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are equivalent.
PubDate: 2018-08-01
DOI: 10.1007/s10485-018-9517-3
Issue No: Vol. 26, No. 4 (2018)

• A Synthetic Version of Lie’s Second Theorem
• Authors: Matthew Burke
Pages: 767 - 798
Abstract: We formulate and prove a generalisation of Lie’s second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with groupoids. Secondly we include groupoids whose underlying spaces are not smooth manifolds. The main intended application is when we replace the category of smooth manifolds with a well-adapted model of synthetic differential geometry. In addition we provide an axiomatic system that provides all the abstract structure that is required to prove Lie’s second theorem.
PubDate: 2018-08-01
DOI: 10.1007/s10485-018-9518-2
Issue No: Vol. 26, No. 4 (2018)

• Fraïssé Limits in Comma Categories
• Authors: Christian Pech; Maja Pech
Pages: 799 - 820
Abstract: Fraïssé’s theorem characterizing the existence of universal homogeneous structures is a cornerstone of model theory. A categorical version of these results was developed by Droste and Göbel. Such an abstract version of Fraïssé theory allows to construct unusual objects that are far away from the usual structures. In this paper we are going to derive sufficient conditions for a comma category to contain universal homogeneous objects. Using this criterion, we characterize homogeneous structures that possess universal homogeneous endomorphisms. The existence of such endomorphisms helps to reduce questions about the full endomorphism monoid to the self-embedding monoid of the structure. As another application we characterize the retracts of homogeneous structures that are induced by universal homogeneous retractions. This extends previous results by Bonato, Delić, Mudrinski, Dolinka, and Kubiś.
PubDate: 2018-08-01
DOI: 10.1007/s10485-018-9519-1
Issue No: Vol. 26, No. 4 (2018)

• On the Maps of Pointfree Topology Which Preserve the Rings of
Integervalued Continuous Functions
• Authors: B. Banaschewski
Pages: 477 - 489
Abstract: This paper establishes various conditions characterizing the homomorphisms $$h: L \rightarrow M$$ of 0-dimensional frames which induce an isomorphism between the rings of all integervalued continuous function, or their bounded parts, on L and M, based on the Lindelöf and the compact coreflection of 0-dimensional frames. This provides natural analogues of familiar results concerning the realvalued continuous functions on completely regular frames, albeit by rather different methods of proof from those originally used in that setting. In addition, it will be shown that the present approach also leads to alternative proofs for the latter.
PubDate: 2018-06-01
DOI: 10.1007/s10485-017-9499-6
Issue No: Vol. 26, No. 3 (2018)

• Crossed Simplicial Group Categorical Nerves
• Authors: Scott Balchin
Pages: 545 - 558
Abstract: We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before looking at twisted versions of some of these nerves. As an application we show how we can use the twisted nerves to give equivariant versions of certain derived stacks.
PubDate: 2018-06-01
DOI: 10.1007/s10485-017-9502-2
Issue No: Vol. 26, No. 3 (2018)

• Bousfield Localisation and Colocalisation of One-Dimensional Model
Structures
• Authors: Scott Balchin; Richard Garner
Abstract: We give an account of Bousfield localisation and colocalisation for one-dimensional model categories—ones enriched over the model category of 0-types. A distinguishing feature of our treatment is that it builds localisations and colocalisations using only the constructions of projective and injective transfer of model structures along right and left adjoint functors, and without any reference to Smith’s theorem.
PubDate: 2018-08-10
DOI: 10.1007/s10485-018-9537-z

• A Multiplication Formula for the Modified Caldero–Chapoton Map
• Authors: David Pescod
Abstract: A frieze in the modern sense is a map from the set of objects of a triangulated category $$\mathsf {C}$$ to some ring. A frieze X is characterised by the property that if $$\tau x\rightarrow y\rightarrow x$$ is an Auslander–Reiten triangle in $$\mathsf {C}$$ , then $$X(\tau x)X(x)-X(y)=1$$ . The canonical example of a frieze is the (original) Caldero–Chapoton map, which send objects of cluster categories to elements of cluster algebras. Holm and Jørgensen (Nagoya Math J 218:101–124, 2015; Bull Sci Math 140:112–131, 2016), the notion of generalised friezes is introduced. A generalised frieze $$X'$$ has the more general property that $$X'(\tau x)X'(x)-X'(y)\in \{0,1\}$$ . The canonical example of a generalised frieze is the modified Caldero–Chapoton map, also introduced in Holm and Jørgensen (2015, 2016). Here, we develop and add to the results in Holm and Jørgensen (2016). We define Condition F for two maps $$\alpha$$ and $$\beta$$ in the modified Caldero–Chapoton map, and in the case when $$\mathsf {C}$$ is 2-Calabi–Yau, we show that it is sufficient to replace a more technical “frieze-like” condition from Holm and Jørgensen (2016). We also prove a multiplication formula for the modified Caldero–Chapoton map, which significantly simplifies its computation in practice.
PubDate: 2018-08-09
DOI: 10.1007/s10485-018-9538-y

• Categories of Locally Hypercompact Spaces and Quasicontinuous Posets
• Authors: Marcel Erné
Abstract: A subset of a topological space is hypercompact if its saturation (the intersection of its neighborhoods) is generated by a finite set. Locally hypercompact spaces are defined by the existence of hypercompact neighborhood bases at each point. We exhibit many useful properties of such spaces, often based on Rudin’s Lemma, which is equivalent to the Ultrafilter Principle and ensures that the Scott spaces of quasicontinuous domains are exactly the locally hypercompact sober spaces. We characterize their patch spaces (the Lawson spaces) as hyperconvex and hyperregular pospaces in which every monotone net has a supremum to which it converges. Moreover, we find extensions to the non-sober case by replacing suprema with cuts, and we provide topological generalizations of known facts for quasicontinuous posets. Similar results are obtained for hypercompactly based spaces and quasialgebraic posets. Furthermore, locally hypercompact spaces are described by certain relations between finite sets and points, providing a quasiuniform approach to such spaces. Our results lead to diverse old and new equivalences and dualities for categories of locally hypercompact spaces or quasicontinuous posets.
PubDate: 2018-08-09
DOI: 10.1007/s10485-018-9536-0

• Categorical Bockstein Sequences
• Authors: Leonid Positselski
Abstract: We construct the reduction of an exact category with a twist functor with respect to an element of its graded center in presence of an exact-conservative forgetful functor annihilating this central element. The construction uses matrix factorizations in a nontraditional way. We obtain the Bockstein long exact sequences for the Ext groups in the exact categories produced by reduction. Our motivation comes from the theory of Artin–Tate motives and motivic sheaves with finite coefficients, and our key techniques generalize those of Positselski (Mosc Math J 11(2):317–402, 2011. arXiv:1006.4343 [math.KT], Section 4).
PubDate: 2018-07-09
DOI: 10.1007/s10485-018-9534-2

• Local Complete Segal Spaces
Abstract: We develop a model structure on bimplicial presheaves on a small site $${\mathscr {C}}$$ , for which the weak equivalences are local (or stalkwise) weak equivalences in the complete Segal model structure. We call this the local Complete Segal model structure. This model structure can be realized as a left Bousfield localization of the Jardine (injective) model structure on the simplicial presheaves on a site $${\mathscr {C}} / {\varDelta }^{op}$$ . Furthermore, it is shown that this model structure is Quillen equivalent to the model structure of the author’s paper (Meadows in TAC 31(24):690–711, 2016). This Quillen equivalence extends an equivalence between the complete Segal space and Joyal model structures, due to Joyal and Tierney (Categories in algebra, geometry and mathematical physics, contemporary mathematics, vol. 431. American Mathematical Society, Providence, pp 277–326, 2007). As an application, we compare the notion of descent in the local Joyal model structure to the notion of descent in the injective model structure. Interestingly, this is a consequence of the Quillen equivalence between the local Joyal and local Complete Segal model structures.
PubDate: 2018-07-07
DOI: 10.1007/s10485-018-9535-1

• Correction to: The Other Closure and Complete Sublocales
• Authors: Maria Manuel Clementino; Jorge Picado; Aleš Pultr
Abstract: In the original publication of the article, the formulation of the c-subfitness condition (c-sfit) in Subsection 5.2 is inaccurate, with effect in Theorem 5.3.
PubDate: 2018-06-29
DOI: 10.1007/s10485-018-9533-3

• Normality, Regularity and Contractive Realvalued Maps
• Authors: E. Colebunders; M. Sioen; W. Van Den Haute
Abstract: For approach spaces normality has been studied from different angles. One way of dealing with it is by focussing on separation by realvalued contractive maps or, equivalently, on Katětov–Tong’s insertion. We call this notion approach normality. Another point of view is using the isomorphism between the category $$\textsf {App}$$ of approach spaces and contractions and the category of lax algebras for the ultrafilter monad and the quantale $$\textsf {P}_{\!\!{_+}}$$ and applying the monoidal definition of normality. We call this notion monoidal normality. Although both normality properties coincide for topological approach spaces, a comparison of both notions for $$\textsf {App}$$ is an open question. In this paper we present a partial solution to this problem. We show that in $$\textsf {App}$$ approach normality implies monoidal normality and that both notions coincide on the subcategory of quasimetric approach spaces. Moreover we investigate the relation between approach normality and regularity. Among other things we prove that approach spaces that are approach normal and regular are uniform.
PubDate: 2018-06-21
DOI: 10.1007/s10485-018-9532-4

• Commutativity in Double Interchange Semigroups
• Authors: Fatemeh Bagherzadeh; Murray Bremner
Abstract: We extend the work of Kock (J Homot Relat Struct 2(2):217–228, 2007) and Bremner and Madariaga (Semigroup Forum 92:335–360,  2016) on commutativity in double interchange semigroups (DIS) to relations with 10 arguments. Our methods involve the free symmetric operad generated by two binary operations with no symmetry, its quotient by the two associative laws, its quotient by the interchange law, and its quotient by all three laws. We also consider the geometric realization of free double interchange magmas by rectangular partitions of the unit square $$I^2$$ . We define morphisms between these operads which allow us to represent elements of free DIS both algebraically as tree monomials and geometrically as rectangular partitions. With these morphisms we reason diagrammatically about free DIS and prove our new commutativity relations.
PubDate: 2018-06-20
DOI: 10.1007/s10485-018-9531-5

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