Authors:Stefan Schröer Pages: 1113 - 1122 Abstract: We show that each sheaf of modules admits a flasque hull, such that homomorphisms into flasque sheaves factor over the flasque hull. On the other hand, we give examples of modules over non-noetherian rings that do not inject into flasque modules. This reveals the impossibility to extend the proof of Serre’s vanishing result for affine schemes with flasque quasicoherent resolutions to the non-noetherian setting. However, we outline how hypercoverings can be used for a reduction to the noetherian case. PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9520-8 Issue No:Vol. 26, No. 6 (2018)

Authors:Arthur J. Parzygnat Pages: 1123 - 1157 Abstract: Given a representation of a \(C^*\) -algebra, thought of as an abstract collection of physical observables, together with a unit vector, one obtains a state on the algebra via restriction. We show that the Gelfand–Naimark–Segal (GNS) construction furnishes a left adjoint of this restriction. To properly formulate this adjoint, it must be viewed as a weak natural transformation, a 1-morphism in a suitable 2-category, rather than as a functor between categories. Weak naturality encodes the functoriality and the universal property of adjunctions encodes the characterizing features of the GNS construction. Mathematical definitions and results are accompanied by physical interpretations. PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9522-6 Issue No:Vol. 26, No. 6 (2018)

Authors:Dikran Dikranjan; Hans-Peter A. Künzi Pages: 1159 - 1184 Abstract: We study cowellpoweredness in the category \(\mathbf{QUnif}\) of quasi-uniform spaces and uniformly continuous maps. A full subcategory \(\mathcal{A}\) of \(\mathbf{QUnif}\) is cowellpowered when the cardinality of the codomains of any class of epimorphisms in \(\mathcal{A}\) , with a fixed common domain, is bounded. We use closure operators in the sense of Dikranjan–Giuli–Tholen which provide a convenient tool for describing the subcategories \(\mathcal{A}\) of \(\mathbf{QUnif}\) and their epimorphisms. Some of the results are obtained by using the knowledge of closure operators, epimorphisms and cowellpoweredness of subcategories of the category \(\mathbf{Top}\) of topological spaces and continuous maps. The transfer is realized by lifting these subcategories along the forgetful functor \(T{:}\mathbf{QUnif}\rightarrow \mathbf{Top}\) and studying when epimorphisms and cowellpoweredness are preserved by the lifting. In other cases closure operators of \(\mathbf{QUnif}\) are used to provide specific results for \(\mathbf{QUnif}\) that have no counterpart in \(\mathbf{Top}\) . This leads to a wealth of cowellpowered categories and a wealth of non-cowellpowered categories of quasi-uniform spaces, in contrast with the current situation in the case of the smaller category \(\mathbf{Unif}\) of uniform spaces, where no example of a non-cowellpowered subcategory is known so far. Finally, we present our main example: a non-cowellpowered full subcategory of \(\mathbf{QUnif}\) which is the intersection of two “symmetric” cowellpowered full subcategories of \(\mathbf{QUnif}\) . PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9523-5 Issue No:Vol. 26, No. 6 (2018)

Authors:Fatemeh Bagherzadeh; Murray Bremner Pages: 1185 - 1210 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: 2018-12-01 DOI: 10.1007/s10485-018-9531-5 Issue No:Vol. 26, No. 6 (2018)

Authors:Leonid Positselski Pages: 1211 - 1263 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: 2018-12-01 DOI: 10.1007/s10485-018-9534-2 Issue No:Vol. 26, No. 6 (2018)

Authors:Nicholas J. Meadows Pages: 1265 - 1281 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: 2018-12-01 DOI: 10.1007/s10485-018-9535-1 Issue No:Vol. 26, No. 6 (2018)

Authors:David Pescod Pages: 1283 - 1304 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: 2018-12-01 DOI: 10.1007/s10485-018-9538-y Issue No:Vol. 26, No. 6 (2018)

Authors:Jorge Picado; Aleš Pultr Pages: 1305 - 1324 Abstract: We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one). PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9542-2 Issue No:Vol. 26, No. 6 (2018)

Authors:Jiří Adámek; Lurdes Sousa Pages: 855 - 872 Abstract: For a functor F whose codomain is a cocomplete, cowellpowered category \(\mathcal {K}\) with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from \(\mathcal {K}(X, F-)\) to \(\mathcal {K}(s, F-)\) form a set. Moreover, the codensity monad has an explicit description using the above natural transformations. Concrete examples are presented, e.g., the codensity monad of the power-set functor \(\mathcal {P}\) assigns to every set X the set of all nonexpanding endofunctions of \(\mathcal {P}X\) . Dually, a set-valued functor F is proved to have a density comonad iff all natural transformations from \(X^F\) to \(2^F\) form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of \({\mathsf {Set}}\) we prove that F has a density comonad iff F is accessible. PubDate: 2018-10-01 DOI: 10.1007/s10485-018-9530-6 Issue No:Vol. 26, No. 5 (2018)

Authors:B. Banaschewski Pages: 873 - 881 Abstract: For the \(\ell \) -ring F(L) introduced in Karimi Feizabadi et al. (Categ Gen Algebr Struct Appl 5:85–102, 2016) and then shown to have an embedding into the familiar \(\ell \) -ring \({\mathfrak R}L\) of all real-valued continuous function on a frame L, the resulting image \({\mathfrak S}L\) in \({\mathfrak R}L\) is characterized here by internal properties within \({\mathfrak R}L\) . Further, a number of results concerning the \({\mathfrak S}L\) are obtained on the basis of this characterization. PubDate: 2018-10-01 DOI: 10.1007/s10485-018-9514-6 Issue No:Vol. 26, No. 5 (2018)

Authors:Maria Manuel Clementino; Jorge Picado; Aleš Pultr Pages: 891 - 906 Abstract: Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or \(T_1\) . Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting. PubDate: 2018-10-01 DOI: 10.1007/s10485-018-9516-4 Issue No:Vol. 26, No. 5 (2018)

Authors:Maria Manuel Clementino; Jorge Picado; Aleš Pultr Pages: 907 - 908 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: 2018-10-01 DOI: 10.1007/s10485-018-9533-3 Issue No:Vol. 26, No. 5 (2018)

Authors:George Janelidze Pages: 1041 - 1046 Abstract: We describe an alternative way of constructing some of the monads, recently introduced by E. Colebunders, R. Lowen, and W. Rosiers for the purposes of categorical topology. PubDate: 2018-10-01 DOI: 10.1007/s10485-018-9525-3 Issue No:Vol. 26, No. 5 (2018)

Authors:George Janelidze; Ross Street Pages: 1065 - 1065 Abstract: In the original publication of the article, Eq. (3.24) was published incorrectly. The corrected equation is given in this correction article. The original article has been corrected. PubDate: 2018-10-01 DOI: 10.1007/s10485-018-9528-0 Issue No:Vol. 26, No. 5 (2018)

Authors:Yasuaki Ogawa Abstract: Given the pair of a dualizing k-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander’s formulas: the first one realizes a module category as a Serre quotient of a suitable functor category. The second one shows a close connection between Auslander–Bridger sequences and recollements. The third one gives a new proof of the higher defect formula which includes the higher Auslander–Reiten duality as a special case. PubDate: 2018-11-13 DOI: 10.1007/s10485-018-9546-y

Authors:Vaino Tuhafeni Shaumbwa Abstract: We show that in an ideal-determined unital category the Higgins commutator can be characterized as the largest binary operation C on subobjects (defined on all subobjects of each object) satisfying the following conditions: (a) C is order-preserving; (b) C(H, K) is always less or equal to the meet of normal closures of H and K; (c) \(C(f(H),f(K))=f(C(H,K))\) for every pair of subobjects H and K of an object X, and every morphism f whose domain is X. PubDate: 2018-11-09 DOI: 10.1007/s10485-018-9548-9

Authors:Jean B. Nganou Abstract: It is proved that the category of extended multisets is dually equivalent to the category of compact Hausdorff MV-algebras with continuous homomorphisms, which is in turn equivalent to the category of complete and completely distributive MV-algebras with homomorphisms that reflect principal maximal ideals. Urysohn–Strauss’s Lemma, Gleason’s Theorem, and projective objects are also investigated for topological MV-algebras. Among other things, it is proved that the only MV-algebras in which Urysohn–Strauss’s Lemma holds are Boolean algebras and that the projective objects in the category of compact Hausdorff MV-algebras are precisely the ones having the 2-element Boolean algebras as factor. PubDate: 2018-11-09 DOI: 10.1007/s10485-018-9547-x

Authors:Behrouz Edalatzadeh Abstract: Let L be a Lie algebra over a field of arbitrary characteristic. In this paper, we give a necessary and sufficient condition for the existence of universal central extensions in the category of crossed modules of Lie algebras over L. Also, we determine the structure of the universal central extension of a crossed L-module and show that the kernel of this extension is related to the first non-abelian homology of L. PubDate: 2018-10-29 DOI: 10.1007/s10485-018-9545-z

Authors:Ivan Kobyzev; Ilya Shapiro Abstract: We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter–Drinfeld contramodules for Hopf algebras. PubDate: 2018-10-15 DOI: 10.1007/s10485-018-9544-0