Authors:Hideto Asashiba Pages: 155 - 186 Abstract: Given a group G, we define suitable 2-categorical structures on the class of all small categories with G-actions and on the class of all small G-graded categories, and prove that 2-categorical extensions of the orbit category construction and of the smash product construction turn out to be 2-equivalences (2-quasi-inverses to each other), which extends the Cohen-Montgomery duality. Further we characterize equivalences in both 2-categories. PubDate: 2017-04-01 DOI: 10.1007/s10485-015-9416-9 Issue No:Vol. 25, No. 2 (2017)

Authors:J. Rosický Pages: 187 - 196 Abstract: We prove that a weak factorization system on a locally presentable category is accessible if and only if it is small generated in the sense of R. Garner. Moreover, we discuss an analogy of Smith’s theorem for accessible model categories. PubDate: 2017-04-01 DOI: 10.1007/s10485-015-9419-6 Issue No:Vol. 25, No. 2 (2017)

Authors:A. R. Shir Ali Nasab; S. N. Hosseini Pages: 197 - 225 Abstract: In this article we give necessary and sufficient conditions for the existence of a pullback of a two sink, in a partial morphism category. PubDate: 2017-04-01 DOI: 10.1007/s10485-015-9420-0 Issue No:Vol. 25, No. 2 (2017)

Authors:Jürgen Fuchs; Gregor Schaumann; Christoph Schweigert Pages: 227 - 268 Abstract: We study a 2-functor that assigns to a bimodule category over a finite \(\Bbbk \) -linear tensor category a \(\Bbbk \) -linear abelian category. This 2-functor can be regarded as a category-valued trace for 1-morphisms in the tricategory of finite tensor categories. It is defined by a universal property that is a categorification of Hochschild homology of bimodules over an algebra. We present several equivalent realizations of this 2-functor and show that it has a coherent cyclic invariance. Our results have applications to categories associated to circles in three-dimensional topological field theories with defects. This is made explicit for the subclass of Dijkgraaf-Witten topological field theories. PubDate: 2017-04-01 DOI: 10.1007/s10485-016-9425-3 Issue No:Vol. 25, No. 2 (2017)

Authors:T. Radul Pages: 269 - 278 Abstract: Let ð”½ be a monad in the category Comp of compact Hausdorff spaces and continuous maps. An abstract convexity was constructed by Radul for each ð”½-algebra of the monad ð”½ in the category Comp. It was proved that if the convexity of the monad ð”½ with some additional properties is binary then ð”½ has good topological properties, in particular, FX is an absolute extensor in the class of 0-dimensional spaces for each openly generated compactum X. We show in this paper that binarity is also a necessary condition. PubDate: 2017-04-01 DOI: 10.1007/s10485-016-9433-3 Issue No:Vol. 25, No. 2 (2017)

Authors:Gabriella Böhm; Stephen Lack Pages: 279 - 301 Abstract: The central object studied in this paper is a multiplier bimonoid in a braided monoidal category \(\mathcal {C}\) , introduced and studied in Böhm and Lack (J. Algebra 423, 853–889 2015). Adapting the philosophy in Janssen and Vercruysse (J. Algebra Appl. 9(2), 275–303 2010), and making some mild assumptions on the category \(\mathcal {C}\) , we introduce a category \(\mathcal {M}\) whose objects are certain semigroups in \(\mathcal {C}\) and whose morphisms A→B can be regarded as suitable multiplicative morphisms from A to the multiplier monoid of B. We equip this category \(\mathcal {M}\) with a monoidal structure and describe multiplier bimonoids in \(\mathcal {C}\) (whose structure morphisms belong to a distinguished class of regular epimorphisms) as certain comonoids in \(\mathcal {M}\) . This provides us with one possible notion of morphism between such multiplier bimonoids. PubDate: 2017-04-01 DOI: 10.1007/s10485-016-9429-z Issue No:Vol. 25, No. 2 (2017)

Authors:Martin Doubek 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: 2017-05-17 DOI: 10.1007/s10485-017-9491-1

Authors:Pedro Nicolás; Manuel Saorín Abstract: Given small dg categories A and B and a B-A-bimodule T, we give necessary and sufficient conditions for the associated derived functors of Hom and the tensor product to be fully faithful. Special emphasis is put on the case when RHom \(_\mathrm{A}\) (T,?) is fully faithful and preserves compact objects, in which case nice recollements situations appear. It is also shown that, given an algebraic compactly generated triangulated category D, all compactly generated co-smashing triangulated subcategories which contain the compact objects appear as the image of such a RHom \(_\mathrm{A}\) (T,?). The results are then applied to the case when A and B are ordinary algebras, comparing the situation with the well-stablished tilting theory of modules. In this way we recover and extend recent results by Bazzoni–Mantese–Tonolo, Chen-Xi and D. Yang. PubDate: 2017-05-15 DOI: 10.1007/s10485-017-9495-x

Authors:Gabriella Böhm; José Gómez-Torrecillas; Stephen Lack 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: 2017-05-11 DOI: 10.1007/s10485-017-9481-3

Authors:Martin Brandenburg Abstract: Let k be a commutative \(\mathbb {Q}\) -algebra. We study families of functors between categories of finitely generated modules which are defined for all commutative k-algebras simultaneously and are compatible with base changes. These operations turn out to be Schur functors associated to k-linear representations of symmetric groups. This result is closely related to Macdonald’s classification of polynomial functors. PubDate: 2017-05-10 DOI: 10.1007/s10485-017-9494-y

Authors:Martín Ortiz-Morales Abstract: Quasi-hereditary algebras were introduced by E. Cline, B. Parshall and L. Scott in order to deal with highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups. These categories have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander, and used in his proof of the first Brauer–Thrall conjecture and later used systematically in his joint work with I. Reiten on stable equivalence, as well as many other applications. Recently, functor categories were used by Martínez-Villa and Solberg to study the Auslander–Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category. We can think of the Auslander–Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category \(\mathrm {Mod}(\mathcal {C} )\) , with \(\mathcal C\) a component of the Auslander–Reiten quiver. PubDate: 2017-05-10 DOI: 10.1007/s10485-017-9493-z

Authors:Bojana Femić 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: 2017-05-08 DOI: 10.1007/s10485-017-9492-0

Authors:David White; Donald Yau 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: 2017-05-02 DOI: 10.1007/s10485-017-9489-8

Authors:Niels Schwartz 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: 2017-04-26 DOI: 10.1007/s10485-017-9488-9

Authors:A. Hager; J. Martinez; C. Monaco 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: 2017-04-20 DOI: 10.1007/s10485-017-9487-x

Authors:Chris Heunen; Vaia Patta Abstract: The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful and having a nearly full Kan extension; there is a functor to the category of geometric lattices, that is nearly full; there are various adjunctions and free constructions on subcategories, inducing a simplification monad; there are two orthogonal factorization systems; some, but not many, combinatorial constructions from matroid theory are functorial. Finally, a characterization of matroids in terms of optimality of the greedy algorithm can be rephrased in terms of limits. PubDate: 2017-04-20 DOI: 10.1007/s10485-017-9490-2

Authors:M. Menni Abstract: We extend the work of Schanuel, Lawvere, Blass and Gates in Objective Number Theory by proving that, for any \({L(X) \in \mathbb {N}[X]}\) , the rig \({\mathbb {N}[X]/(X = L(X))}\) is the Burnside rig of a prextensive category. PubDate: 2017-04-07 DOI: 10.1007/s10485-016-9475-6

Authors:Nelson Martins-Ferreira 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: 2017-03-24 DOI: 10.1007/s10485-017-9486-y

Authors:Javier J. Gutiérrez; Constanze Roitzheim 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: 2017-03-16 DOI: 10.1007/s10485-017-9485-z

Authors:Jan Foniok; Claude Tardif 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: 2017-03-14 DOI: 10.1007/s10485-017-9484-0