Authors:Victoria Lebed; Friedrich Wagemann Pages: 455 - 488 Abstract: Abstract The Yang-Baxter equation plays a fundamental role in various areas of mathematics. Its solutions, called braidings, are built, among others, from Yetter-Drinfel’d modules over a Hopf algebra, from self-distributive structures, and from crossed modules of groups. In the present paper these three sources of solutions are unified inside the framework of Yetter-Drinfel’d modules over a braided system. A systematic construction of braiding structures on such modules is provided. Some general categorical methods of obtaining such generalized Yetter-Drinfel’d (=GYD) modules are described. Among the braidings recovered using these constructions are the Woronowicz and the Hennings braidings on a Hopf algebra. We also introduce the notions of crossed modules of shelves / Leibniz algebras, and interpret them as GYD modules. This yields new sources of braidings. We discuss whether these braidings stem from a braided monoidal category, and discover several non-strict pre-tensor categories with interesting associators. PubDate: 2017-08-01 DOI: 10.1007/s10485-015-9421-z Issue No:Vol. 25, No. 4 (2017)

Authors:Vasily A. Dolgushev; Christopher L. Rogers Pages: 489 - 503 Abstract: Abstract We construct a symmetric monoidal category \({\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}\) whose objects are shifted L ∞ -algebras equipped with a complete descending filtration. Morphisms of this category are “enhanced” infinity morphisms between shifted L ∞ -algebras. We prove that any category enriched over \({\mathfrak {S}}\mathsf {Lie}_{\infty }^{\text {MC}}\) can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L ∞ -algebra point of view. We use this construction in a subsequent paper (Dolgushev et al. Adv. Math. 274, 562–605, 2015) to produce a simplicial model of a (∞,1)-category whose objects are homotopy algebras of a fixed type. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9424-4 Issue No:Vol. 25, No. 4 (2017)

Authors:Jun Pei; Chengming Bai; Li Guo Pages: 505 - 538 Abstract: Abstract This paper establishes a procedure that splits the operations in any algebraic operad, generalizing previous notions of splitting algebraic structures, from the dendriform algebra of Loday splitting the associative operation to the successors splitting binary operads. The separately treated bisuccessor and trisuccessor for binary operads are unified for general operads through the notion of configuration. Applications are provided for various n-algebras, the \(A_{\infty }\) and \(L_{\infty }\) algebras. Further, the concept of a Rota-Baxter operator, first showing its importance in the associative and Lie algebra contexts and then generalized to binary operads, is defined for all operads. The well-known connection from Rota-Baxter operators to dendriform algebras and its numerous extensions are expanded as the link from (relative) Rota-Baxter operators on operads to splittings of the operads PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9431-5 Issue No:Vol. 25, No. 4 (2017)

Authors:Xiao-Wu Chen Pages: 539 - 568 Abstract: Abstract For an additive category with a Serre duality and a finite group action, we compute explicitly the Serre duality on the category of equivariant objects. We prove that under certain conditions, the equivarianzation of an additive category with a periodic Serre duality still has a periodic Serre duality. A similar result is proved for fractionally Calabi-Yau triangulated categories. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9432-4 Issue No:Vol. 25, No. 4 (2017)

Authors:Wiesław Kubiś; Dragan Mašulović Pages: 569 - 602 Abstract: Abstract We develop a theory of Katětov functors which provide a uniform way of constructing Fraïssé limits. Among applications, we present short proofs and improvements of several recent results on the structure of the group of automorphisms and the semigroup of endomorphisms of some Fraïssé limits. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9461-z Issue No:Vol. 25, No. 4 (2017)

Authors:John Bourke; Nick Gurski Pages: 603 - 624 Abstract: Abstract We discuss the folklore construction of the Gray tensor product of 2-categories as obtained by factoring the map from the funny tensor product to the cartesian product. We show that this factorisation can be obtained without using a concrete presentation of the Gray tensor product, but merely its defining universal property, and use it to give another proof that the Gray tensor product forms part of a symmetric monoidal structure. The main technical tool is a method of producing new algebra structures over Lawvere 2-theories from old ones via a factorisation system. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9467-6 Issue No:Vol. 25, No. 4 (2017)

Authors:Francesco Genovese Pages: 625 - 657 Abstract: Abstract We study right quasi-representable differential graded bimodules as quasi-functors between dg-categories. We prove that a quasi-functor has a left adjoint if and only if it is left quasi-representable. With this characterisation, we prove an existence result of adjoints, under suitable hypotheses on the dg-categories. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9470-y Issue No:Vol. 25, No. 4 (2017)

Authors:Xabier García-Martínez Pages: 659 - 661 Abstract: Abstract We prove that a monoid M is a group if and only if, in the category of monoids, all points over M are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden (Pré-Publicações DMUC 16–21, 1–41 2016) which characterises groups amongst monoids as the protomodular objects. PubDate: 2017-08-01 DOI: 10.1007/s10485-016-9471-x Issue No:Vol. 25, No. 4 (2017)

Authors:M. Menni Pages: 663 - 707 Abstract: 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: 2017-08-01 DOI: 10.1007/s10485-016-9475-6 Issue No:Vol. 25, No. 4 (2017)

Authors:John Frith; Anneliese Schauerte Pages: 303 - 321 Abstract: Abstract Defining objects using generators and relations has seen substantial application in the theory of frames. It is the aim of this paper to establish such a technique for partial frames, thus making it available in a variety of contexts. A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets in question are specified by means of a so-called selection function. The theory is general enough to include, as examples, bounded distributive lattices, σ-frames, κ-frames and indeed frames, but a small collection of elementary axioms suffices to describe the selection functions and thus the designated subsets. In this paper we are concerned with establishing techniques for constructing objects given certain generators and the relations that they should satisfy. Our method involves embedding the generators in an appropriate meet-semilattice, moving to the free partial frame over that meet-semilattice, and then using the relations to form a quotient with the required joins. We use a modification of Johnstone’s coverages on meet-semilattices [12] to construct partial frames freely generated by sites. We conclude with a number of applications, including the construction of coproducts for partial frames and a general method for freely adjoining complements. PubDate: 2017-06-01 DOI: 10.1007/s10485-015-9417-8 Issue No:Vol. 25, No. 3 (2017)

Authors:Krzysztof Kapulkin; Karol Szumiło Pages: 323 - 347 Abstract: Abstract We show that the quasicategory of frames of a cofibration category, introduced by the second-named author, is equivalent to its simplicial localization. PubDate: 2017-06-01 DOI: 10.1007/s10485-015-9422-y Issue No:Vol. 25, No. 3 (2017)

Authors:H. Boustique; G. Richardson Pages: 349 - 361 Abstract: Abstract Convergence approach spaces, defined by E. Lowen and R. Lowen, possess both quantitative and topological properties. These spaces are equipped with a structure which provides information as to whether or not a sequence or filter approximately converges. P. Brock and D. Kent showed that the category of convergence approach spaces with contractions as morphisms is isomorphic to the category of limit tower spaces. It is shown below that every limit tower space has a compactification. Moreover, a characterization of the limit tower spaces which possess a strongly regular compactification is given here. Further, a strongly regular S-compactification of a limit tower space is studied, where S is a limit tower monoid acting on the limit tower space. PubDate: 2017-06-01 DOI: 10.1007/s10485-016-9426-2 Issue No:Vol. 25, No. 3 (2017)

Authors:Guram Bezhanishvili; Nick Bezhanishvili; Sumit Sourabh; Yde Venema Pages: 381 - 401 Abstract: Abstract By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras. PubDate: 2017-06-01 DOI: 10.1007/s10485-016-9434-2 Issue No:Vol. 25, No. 3 (2017)

Authors:B. A. Davey; M. Haviar; H. A. Priestley Pages: 403 - 430 Abstract: This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set. The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same; in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures. Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes—often varieties—are plentiful and varied, and in many cases the associated topological prevariety is standard. PubDate: 2017-06-01 DOI: 10.1007/s10485-016-9436-0 Issue No:Vol. 25, No. 3 (2017)

Authors:A. Razafindrakoto; D. Holgate Pages: 431 - 445 Abstract: Abstract Viewing neighbourhood operators as lax natural transformations introduces an efficiency in calculations and proofs and suggests further applications. To highlight the advantages of this approach, classes of open, closed, initial and final morphisms are studied. In addition new proofs are given to previous results and a new example that departs from the current factorisation system paradigm is exhibited. PubDate: 2017-06-01 DOI: 10.1007/s10485-016-9441-3 Issue No:Vol. 25, No. 3 (2017)

Authors:Simeon Pol’shin Pages: 447 - 453 Abstract: Abstract We construct relative abelian categories in the sense of MacLane for models of algebraic systems in (co)complete abelian categories. As an example, we consider an analogue of Hochschild-Mitchell cohomology for the functor of Yoneda embedding. PubDate: 2017-06-01 DOI: 10.1007/s10485-016-9466-7 Issue No:Vol. 25, No. 3 (2017)

Authors:B. Banaschewski Abstract: Abstract This paper establishes various conditions characterizing the homomorphisms \(h: L \rightarrow M\) of 0-dimensional frames which induce an isomorphism between the rings of all integervalued continuous function, or their bounded parts, on L and M, based on the Lindelöf and the compact coreflection of 0-dimensional frames. This provides natural analogues of familiar results concerning the realvalued continuous functions on completely regular frames, albeit by rather different methods of proof from those originally used in that setting. In addition, it will be shown that the present approach also leads to alternative proofs for the latter. PubDate: 2017-07-17 DOI: 10.1007/s10485-017-9499-6

Authors:Dragan Mašulović Abstract: Abstract In this paper we give a new proof of the Nešetřil–Rödl Theorem, a deep result of discrete mathematics which is one of the cornerstones of the structural Ramsey theory. In contrast to the well-known proofs which employ intricate combinatorial strategies, this proof is spelled out in the language of category theory and the main result follows by applying several simple categorical constructions. The gain from the approach we present here is that, instead of giving the proof in the form of a large combinatorial construction, we can start from a few building blocks and then combine them into the final proof using general principles. PubDate: 2017-07-15 DOI: 10.1007/s10485-017-9500-4

Authors:Oghenetega Ighedo Abstract: Abstract An ideal I of a commutative ring A with identity is called a z-ideal if whenever two elements of A belong to the same maximal ideals and one of the elements is in I, then so is the other. For a completely regular frame L we denote by \({{\mathrm{ZId}}}(\mathcal {R}L)\) the lattice of z-ideals of the ring \(\mathcal {R}L\) of continuous real-valued functions on L. It is a coherent frame, and it is known that \(L\mapsto {{\mathrm{ZId}}}(\mathcal {R}L)\) is the object part of a functor \(\mathsf {Z}:\mathbf {CRFrm}\rightarrow \mathbf {CohFrm}\) , where \(\mathbf {CRFrm}\) is the category of completely regular frames and frame homomorphisms, and \(\mathbf {CohFrm}\) is the category of coherent frames and coherent maps. We explore when this functor preserves and reflects the property of being a Heyting homomorphism, and also when it preserves and reflects the variants of openness of Banaschewski and Pultr (Appl Categ Struct 2:331–350, 1994). We also record some other properties of this functor that have hitherto not been stated anywhere. PubDate: 2017-06-21 DOI: 10.1007/s10485-017-9498-7

Authors:Marcelo Aguiar; Mariana Haim; Ignacio López Franco Abstract: Abstract We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of n-monoidal categories and discuss several naturally occurring examples in which \(n\le 3\) . PubDate: 2017-06-05 DOI: 10.1007/s10485-017-9497-8