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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

Authors:Oghenetega Ighedo Pages: 459 - 476 Abstract: An ideal I of a commutative ring A with identity is called a z-ideal if whenever two elements of A belong to the same maximal ideals and one of the elements is in I, then so is the other. For a completely regular frame L we denote by \({{\mathrm{ZId}}}(\mathcal {R}L)\) the lattice of z-ideals of the ring \(\mathcal {R}L\) of continuous real-valued functions on L. It is a coherent frame, and it is known that \(L\mapsto {{\mathrm{ZId}}}(\mathcal {R}L)\) is the object part of a functor \(\mathsf {Z}:\mathbf {CRFrm}\rightarrow \mathbf {CohFrm}\) , where \(\mathbf {CRFrm}\) is the category of completely regular frames and frame homomorphisms, and \(\mathbf {CohFrm}\) is the category of coherent frames and coherent maps. We explore when this functor preserves and reflects the property of being a Heyting homomorphism, and also when it preserves and reflects the variants of openness of Banaschewski and Pultr (Appl Categ Struct 2:331–350, 1994). We also record some other properties of this functor that have hitherto not been stated anywhere. PubDate: 2018-06-01 DOI: 10.1007/s10485-017-9498-7 Issue No:Vol. 26, No. 3 (2018)

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)

Authors:Hiroyuki Nakaoka Pages: 491 - 544 Abstract: In this article, we introduce the notion of concentric twin cotorsion pair on a triangulated category. This notion contains the notions of t-structure, cluster tilting subcategory, co-t-structure and functorally finite rigid subcategory as examples. Moreover, a recollement of triangulated categories can be regarded as a special case of concentric twin cotorsion pair. To any concentric twin cotorsion pair, we associate a pretriangulated subquotient category. This enables us to give a simultaneous generalization of the Iyama–Yoshino reduction and the recollement of cotorsion pairs. This allows us to give a generalized mutation on cotorsion pairs defined by the concentric twin cotorsion pair. PubDate: 2018-06-01 DOI: 10.1007/s10485-017-9501-3 Issue No:Vol. 26, No. 3 (2018)

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)

Authors:Rory B. B. Lucyshyn-Wright Pages: 559 - 596 Abstract: We define and study a notion of commutant for \(\mathscr {V}\) -enriched \({\mathscr {J}}\) -algebraic theories for a system of arities \({\mathscr {J}}\) , recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith’s notion of commutant for Lawvere theories as well as a notion of commutant for \(\mathscr {V}\) -monads on a symmetric monoidal closed category \(\mathscr {V}\) . This entails a thorough study of commutation and Kronecker products of operations in \({\mathscr {J}}\) -theories. In view of the equivalence between \({\mathscr {J}}\) -theories and \({\mathscr {J}}\) -ary monads we reconcile this notion of commutation with Kock’s notion of commutation of cospans of monads and, in particular, the notion of commutative monad. We obtain notions of \({\mathscr {J}}\) -ary commutant and absolute commutant for \({\mathscr {J}}\) -ary monads, and we show that for finitary monads on \(\text {Set}\) the resulting notions of finitary commutant and absolute commutant coincide. We examine the relation of the notion of commutant to both the notion of codensity monad and the notion of algebraic structure in the sense of Lawvere. PubDate: 2018-06-01 DOI: 10.1007/s10485-017-9503-1 Issue No:Vol. 26, No. 3 (2018)

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

Authors:Nicholas J. Meadows 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

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

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

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