Authors:Tomáš Jakl; Achim Jung; Aleš Pultr Abstract: It is shown that every d-frame admits a complete lattice of quotients. Quotienting may be triggered by a binary relation on one of the two constituent frames, or by changes to the consistency or totality structure, but as these are linked by the reasonableness conditions of d-frames, the result in general will be that both frames are factored and both consistency and totality are increased. PubDate: 2019-01-05 DOI: 10.1007/s10485-018-09553-7

Authors:Sondre Kvamme; René Marczinzik Abstract: We review the theory of Co-Gorenstein algebras, which was introduced in Beligiannis (Commun Algebra 28(10):4547–4596, 2000). We show a connection between Co-Gorenstein algebras and the Nakayama and Generalized Nakayama conjecture. PubDate: 2019-01-04 DOI: 10.1007/s10485-018-09554-6

Authors:Daniel Lin Abstract: Just as the presheaf category is the free cocompletion of any small category, there is an analogous notion of free cocompletion for any small restriction category. In this paper, we extend the work on restriction presheaves to presheaves over join restriction categories, and show that the join restriction category of join restriction presheaves is equivalent to some partial map category of sheaves. We then use this to show that the Yoneda embedding exhibits the category of join restriction presheaves as the free cocompletion of any small join restriction category. PubDate: 2019-01-04 DOI: 10.1007/s10485-018-09555-5

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)

Abstract: We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and \(\varDelta \) -complexes, respectively. The functors of classifying spaces and face posets are compatible with these homotopy theories. In contrast with the classical settings of finite spaces and simplicial complexes, the universality of morphisms and simplices plays a central role in this paper. PubDate: 2018-12-01 DOI: 10.1007/s10485-018-9552-0

Authors:Ahmet A. Husainov Abstract: The paper is devoted to homology groups of cubical sets with coefficients in contravariant systems of abelian groups. The study is based on the proof of the assertion that the homology groups of the category of cubes with coefficients in the diagram of abelian groups are isomorphic to the homology groups of the normalized complex of the cubical abelian group corresponding to this diagram. The main result shows that the homology groups of a cubical set with coefficients in a contravariant system of abelian groups are isomorphic to the values of left derived functors of the colimit functor on this contravariant system. This is used to obtain the isomorphism criterion for homology groups of cubical sets with coefficients in contravariant systems, and also to construct spectral sequences for the covering of a cubical set and for a morphism between cubical sets. The use of the main result leads to an isomorphism between homology groups of small category with coefficients in the diagram of abelian groups and normalized homology groups of its cubical nerve with coefficients in the corresponding contravariant system. PubDate: 2018-11-22 DOI: 10.1007/s10485-018-9550-2

Authors:Krzysztof Ziemiański Abstract: In this paper, we introduce the notions of stable future, past and total component systems on a directed space with no loops. Then, we associate the stable component category to a stable (future, past or total) component system. Stable component categories are enriched in some monoidal category, eg. the homotopy category of spaces, and carry information about the spaces of directed paths between particular points. It is shown that the geometric realizations of finite pre-cubical sets with no loops admit unique minimal stable (future/past/total) component systems. These constructions provide a new family of invariants for directed spaces. PubDate: 2018-11-21 DOI: 10.1007/s10485-018-9551-1

Authors:Xuexing Lu; Yu Ye; Sen Hu Abstract: Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to formalize this graphical calculus, a topological one based on the notion of a processive plane graph and a combinatorial one based on the notion of a planarly ordered processive graph, which serves as a combinatorial counterpart of a deformation class of processive plane graphs. We demonstrate the equivalence of Joyal and Street’s graphical calculus and the theory of upward planar drawings. We introduce the category of semi-tensor schemes, and give a construction of a free monoidal category on a semi-tensor scheme. We deduce the unit convention as a kind of quotient construction, and show an idea to generalize the unit convention. Finally, we clarify the relation of the unit convention and Joyal and Street’s construction of a free monoidal category on a tensor scheme. PubDate: 2018-11-19 DOI: 10.1007/s10485-018-9549-8

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