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:Guram Bezhanishvili; Patrick J. Morandi Pages: 1077 - 1095 Abstract: In our previous paper, in order to develop the pointfree theory of compactifications of ordered spaces, we introduced the concept of a proximity on a biframe as a generalization of the concept of a strong inclusion on a biframe. As a natural next step, we introduce the concept of a proximity morphism between proximity biframes. Like in the case of de Vries algebras and proximity frames, we show that the proximity biframes and proximity morphisms between them form a category PrBFrm in which composition is not function composition. We prove that the category KRBFrm of compact regular biframes and biframe homomorphisms is a proper full subcategory of PrBFrm that is equivalent to PrBFrm. We also show that PrBFrm is equivalent to the category PrFrm of proximity frames, and give a simple description of the concept of regularization using the language of proximity biframes. Finally, we describe the dual equivalence of PrBFrm and the category Nach of Nachbin spaces, which provides a direct way to construct compactifications of ordered spaces. PubDate: 2017-12-01 DOI: 10.1007/s10485-016-9476-5 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:Nelson Martins-Ferreira Pages: 1137 - 1158 Abstract: A detailed description of a normalized internal bicategory in the category of groups is derived from the general description of internal bicategories in weakly Mal’tsev categories endowed with a V-Mal’tsev operation in the sense of Pedicchio. The example of bicategory of paths in a topological abelian group is presented. PubDate: 2017-12-01 DOI: 10.1007/s10485-017-9486-y 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:Jim Andrianopoulos Pages: 863 - 873 Abstract: This article shows that the axioms of a skew monoidal category are independent and that its unit is unique up to a unique isomorphism together with an analogue of this result for monoidal functors between skew monoidal categories. It is also noted that these results carry over to skew monoidales before some benefits of certain extra structure on the unit maps of a skew monoidal category are discussed. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9468-5 Issue No:Vol. 25, No. 5 (2017)

Authors:Driss Bennis; J. R. García Rozas; Luis Oyonarte Pages: 907 - 915 Abstract: In this paper we are interested in studying the stability question of subcategories of an abelian category \(\mathcal {A}\) constituted of all objects that admit (proper) coproper resolutions (resp. (coproper) proper coresolutions) with terms in a subcategory \(\mathcal {E}\) of \(\mathcal {A}\) . Using a new approach, we give an affirmative answer to the stability question on these categories under the condition that \(\mathcal {E}\) is closed under finite direct sums. This result generalizes Huang’s affirmative answer to the well-known stability question of Gorenstein categories raised by Sather-Wagstaff, Sharif and White. We end the paper with an example showing that the condition imposed on \(\mathcal {E}\) cannot be dropped. PubDate: 2017-10-01 DOI: 10.1007/s10485-016-9478-3 Issue No:Vol. 25, No. 5 (2017)

Authors:Graham Manuell 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: 2017-11-09 DOI: 10.1007/s10485-017-9507-x

Authors:Manuel Cortés-Izurdiaga; Alberto Facchini 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: 2017-10-30 DOI: 10.1007/s10485-017-9505-z

Authors:Pedro Resende 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: 2017-10-27 DOI: 10.1007/s10485-017-9506-y

Authors:Rory B. B. Lucyshyn-Wright 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: 2017-10-09 DOI: 10.1007/s10485-017-9503-1

Authors:Alexander Campbell Abstract: This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the morphisms of the underlying category. This is the natural setting in which to introduce the notion of locally weak comonad, which is fundamental to the theory of enriched algebraic weak factorisation systems. The equivalence, for a monoidal closed category \(\mathcal {V}\) , between tensored \(\mathcal {V}\) -categories and hommed \(\mathcal {V}\) -actegories is extended to the skew setting and easily proved by recognising both skew \(\mathcal {V}\) -categories and skew \(\mathcal {V}\) -actegories as equivalent to special kinds of skew \(\mathcal {V}\) -proactegory. PubDate: 2017-10-05 DOI: 10.1007/s10485-017-9504-0