Authors:Jan Grebík Pages: 1 - 6 Abstract: We disprove a conjecture from Kubiś and Mašulović [2] by showing the existence of a Fraïssé class \(\mathcal {C}\) which does not admit a Katětov functor. On the other hand, we show that the automorphism group of the Fraïssé limit of \(\mathcal {C}\) is universal, as it happens in the presence of a Katětov functor. PubDate: 2018-02-01 DOI: 10.1007/s10485-016-9469-4 Issue No:Vol. 26, No. 1 (2018)

Authors:Mehmet Akif Erdal; Özgün Ünlü Pages: 7 - 28 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: 2018-02-01 DOI: 10.1007/s10485-016-9477-4 Issue No:Vol. 26, No. 1 (2018)

Authors:S. Gorchinskiy; V. Guletskiĭ Pages: 29 - 46 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: 2018-02-01 DOI: 10.1007/s10485-016-9480-9 Issue No:Vol. 26, No. 1 (2018)

Authors:Gabriella Böhm; José Gómez-Torrecillas; Stephen Lack Pages: 47 - 111 Abstract: Based on the novel notion of ‘weakly counital fusion morphism’, regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields (Böhm et al. Trans. Amer. Math. Soc. 367, 8681–8721, 2015) and multiplier bimonoids in braided monoidal categories (Böhm and Lack, J. Algebra 423, 853–889, 2015). Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations. PubDate: 2018-02-01 DOI: 10.1007/s10485-017-9481-3 Issue No:Vol. 26, No. 1 (2018)

Authors:Jan Foniok; Claude Tardif Pages: 113 - 128 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: 2018-02-01 DOI: 10.1007/s10485-017-9484-0 Issue No:Vol. 26, No. 1 (2018)

Authors:A. Hager; J. Martinez; C. Monaco Pages: 129 - 151 Abstract: This paper explicates some basic categorical ideas in the category of the title, W ∗ (e.g., products and coproducts, monics, epics, and extremal monics, …) for the record, and for immediate application to description of some epireflective subcategories generated in various ways (at least six) by subobjects E of the reals \(\mathbb {R}\) . These E have a very special place in W ∗ because of the Yosida Representation G ≤ C(Y G) which says directly that \(\mathbb {R}\) is a co-separator in W ∗, and implies less directly that G ≤ C(Y G) is the epicomplete monoreflection of G. The E are exactly the nonterminal quasi-initial objects of W ∗ and generate the atoms in the lattice of epireflective subcategories of W ∗. PubDate: 2018-02-01 DOI: 10.1007/s10485-017-9487-x Issue No:Vol. 26, No. 1 (2018)

Authors:David White; Donald Yau Pages: 153 - 203 Abstract: We provide a very general approach to placing model structures and semi-model structures on algebras over symmetric colored operads. Our results require minimal hypotheses on the underlying model category \(\mathcal {M}\) , and these hypotheses vary depending on what is known about the colored operads in question. We obtain results for the classes of colored operad which are cofibrant as a symmetric collection, entrywise cofibrant, or arbitrary. As the hypothesis on the operad is weakened, the hypotheses on \(\mathcal {M}\) must be strengthened. Via a careful development of the categorical algebra of colored operads we provide a unified framework which allows us to build (semi-)model structures for all three of these classes of colored operads. We then apply these results to provide conditions on \(\mathcal {M}\) , on the colored operad O, and on a class \(\mathcal {C}\) of morphisms in \(\mathcal {M}\) so that the left Bousfield localization of \(\mathcal {M}\) with respect to \(\mathcal {C}\) preserves O-algebras. Even the strongest version of our hypotheses on \(\mathcal {M}\) is satisfied for model structures on simplicial sets, chain complexes over a field of characteristic zero, and symmetric spectra. We obtain results in these settings allowing us to place model structures on algebras over any colored operad, and to conclude that monoidal Bousfield localizations preserve such algebras. PubDate: 2018-02-01 DOI: 10.1007/s10485-017-9489-8 Issue No:Vol. 26, No. 1 (2018)

Authors:Julia Ramos González 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 [15] 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 [12] into a bi-monoidal structure on \(\mathsf {Grt}_k\) . PubDate: 2018-01-12 DOI: 10.1007/s10485-017-9511-1

Authors:Pavel Etingof Pages: 965 - 969 Abstract: Let G be a finite group. There is a standard theorem on the classification of G-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of G). Namely, such an algebra is of the form A=Fun H (G,B), where H is a subgroup of G, and B is a simple algebra of the corresponding type with an H-action. We explain that such a result holds in the generality of algebras over a linear operad. This allows one to extend Theorem 5.5 of Sciarappa (arXiv:1506.07565) on the classification of simple commutative algebras in the Deligne category Rep(S t ) to algebras over any finitely generated linear operad. PubDate: 2017-12-01 DOI: 10.1007/s10485-016-9435-1 Issue No:Vol. 25, No. 6 (2017)

Authors:Bing Wang; Yuan Yao; Yu Ye Pages: 971 - 990 Abstract: This paper is motivated by the observation that the characteristic morphism of an algebra relates to certain smoothness condition closely. We show that for an algebra A of finite global dimension, if the characteristic morphism is injective, then A has finite Hochschild cohomology dimension. In particular, if A is semi-simple, then the characteristic morphism is injective if and only if A is homologically smooth. Moreover, the characteristic morphism of a finite dimensional path algebra is injective. Recall that a path algebra is always homologically smooth. PubDate: 2017-12-01 DOI: 10.1007/s10485-016-9437-z Issue No:Vol. 25, No. 6 (2017)

Authors:Volodymyr Lyubashenko Pages: 991 - 1036 Abstract: We describe the category of homotopy coalgebras, concentrating on properties of relatively cofree homotopy coalgebras, morphisms and coderivations from an ordinary coalgebra to a relatively cofree homotopy coalgebra, morphisms and coderivations between coalgebras of latter type. Cobar- and bar-constructions between counit-complemented curved coalgebras, unit-complemented curved algebras and curved homotopy coalgebras are described. Using twisting cochains an adjunction between cobar- and bar-constructions is derived under additional assumptions. PubDate: 2017-12-01 DOI: 10.1007/s10485-016-9440-4 Issue No:Vol. 25, No. 6 (2017)

Authors:Amartya Goswami; Zurab Janelidze Pages: 1037 - 1043 Abstract: A quasi-pointed category in the sense of D. Bourn is a finitely complete category \(\mathcal {C}\) having an initial object such that the unique morphism from the initial object to the terminal object is a monomorphism. When instead this morphism is an isomorphism, we obtain a (finitely complete) pointed category, and as it is well known, the structure of zero morphisms in a pointed category determines an enrichment of the category in the category of pointed sets. In this note we examine quasi-pointed categories through the structure formed by the zero morphisms (i.e. the morphisms which factor through the initial object), with the aim to compare this structure with an enrichment in the category of pointed sets. PubDate: 2017-12-01 DOI: 10.1007/s10485-016-9462-y Issue No:Vol. 25, No. 6 (2017)

Authors:J. Bruno; P. Szeptycki Pages: 1045 - 1058 Abstract: Premetrics and premetrisable spaces have been long studied and their topological interrelationships are well-understood. Consider the category Pre of premetric spaces and ðœ– − δ continuous functions as morphisms. The absence of the triangle inequality implies that the faithful functor Pre→Top - where a premetric space is sent to the topological space it generates - is not full. Moreover, the sequential nature of topological spaces generated from objects in Pre indicates that this functor is not surjective on objects either. Developed from work by Flagg and Weiss, we illustrate an extension Pre↪P together with a faithful and surjective on objects left adjoint functor P→Top as an extension of Pre→Top. We show this represents an optimal scenario given that Pre→Top preserves coproducts only. The objects in P are metric-like objects valued on value distributive lattices whose limits and colimits we show to be generated by free locales on discrete sets. PubDate: 2017-12-01 DOI: 10.1007/s10485-016-9465-8 Issue No:Vol. 25, No. 6 (2017)

Authors:Rafael Fernández-Casado; Xabier García-Martínez; Manuel Ladra Pages: 1059 - 1076 Abstract: The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general, so the technique used in the proof for the Lie case cannot be applied. Finally we move on to the framework of the Loday-Pirashvili category \(\mathcal {LM}\) in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case. PubDate: 2017-12-01 DOI: 10.1007/s10485-016-9472-9 Issue No:Vol. 25, No. 6 (2017)

Authors:Themba Dube Pages: 1097 - 1111 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: 2017-12-01 DOI: 10.1007/s10485-016-9479-2 Issue No:Vol. 25, No. 6 (2017)

Authors:Javier J. Gutiérrez; Constanze Roitzheim Pages: 1113 - 1136 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: 2017-12-01 DOI: 10.1007/s10485-017-9485-z Issue No:Vol. 25, No. 6 (2017)

Authors:Niels Schwartz Pages: 1159 - 1185 Abstract: The category Spec of spectral spaces is a reflective subcategory of the category Top of topological spaces. We compare properties of topological spaces, or of continuous maps between topological spaces, with properties of their spectral reflections. It is shown that several classical constructions with topological spaces can be produced using spectral reflections. PubDate: 2017-12-01 DOI: 10.1007/s10485-017-9488-9 Issue No:Vol. 25, No. 6 (2017)

Authors:Martin Doubek Pages: 1187 - 1198 Abstract: We give a direct combinatorial proof that the modular envelope of the cyclic operad \(\mathcal {A} ss \) is the modular operad of (the homeomorphism classes of) 2D compact surfaces with boundary with marked points. PubDate: 2017-12-01 DOI: 10.1007/s10485-017-9491-1 Issue No:Vol. 25, No. 6 (2017)

Authors:Bojana Femić Pages: 1199 - 1228 Abstract: We prove that if a finite tensor category \({\mathcal C}\) is symmetric, then the monoidal category of one-sided \({\mathcal C}\) -bimodule categories is symmetric. Consequently, the Picard group of \({\mathcal C}\) (the subgroup of the Brauer–Picard group introduced by Etingov–Nikshych–Gelaki) is abelian in this case. We then introduce a cohomology over such \({\mathcal C}\) . An important piece of tool for this construction is the computation of dual objects for bimodule categories and the fact that for invertible one-sided \({\mathcal C}\) -bimodule categories the evaluation functor involved is an equivalence, being the coevaluation functor its quasi-inverse, as we show. Finally, we construct an infinite exact sequence à la Villamayor–Zelinsky for \({\mathcal C}\) . It consists of the corresponding cohomology groups evaluated at three types of coefficients which repeat periodically in the sequence. We compute some subgroups of the groups appearing in the sequence. PubDate: 2017-12-01 DOI: 10.1007/s10485-017-9492-0 Issue No:Vol. 25, No. 6 (2017)

Authors:J. Rosický; W. Tholen 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: 2017-12-19 DOI: 10.1007/s10485-017-9510-2