Authors:John Frith; Anneliese Schauerte Pages: 303 - 321 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: 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: 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:Christopher Townsend Pages: 363 - 380 Abstract: Given a particular collection of categorical axioms, aimed at capturing properties of the category of locales, we show that if \(\mathcal {C}\) is a category that satisfies the axioms then so too is the category \([ G, \mathcal {C}]\) of G-objects, for any internal group G. To achieve this we prove a general categorical result: if an object S is double exponentiable in a category with finite products then so is its associated trivial G-object (S, π 2:G × S → S). The result holds even if S is not exponentiable. An example is given of a category \(\mathcal {C}\) that satisfies the axioms, but for which there is no elementary topos \(\mathcal {E}\) such that \(\mathcal {C}\) is the category of locales over \(\mathcal {E}\) . It is shown, in outline, how the results can be extended from groups to groupoids. PubDate: 2017-06-01 DOI: 10.1007/s10485-016-9430-6 Issue No:Vol. 25, No. 3 (2017)

Authors:Guram Bezhanishvili; Nick Bezhanishvili; Sumit Sourabh; Yde Venema Pages: 381 - 401 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: 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: 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)

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

Authors:Marcelo Aguiar; Mariana Haim; Ignacio López Franco 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

Authors:Rory B. B. Lucyshyn-Wright Abstract: Certain axiomatic notions of affine space over a ring and convex space over a preordered ring are examples of the notion of \(\mathscr {T}\) -algebra for an algebraic theory \(\mathscr {T}\) in the sense of Lawvere. Herein we study the notion of commutant for Lawvere theories that was defined by Wraith and generalizes the notion of centralizer clone. We focus on the Lawvere theory of left R -affine spaces for a ring or rig R, proving that this theory can be described as a commutant of the theory of pointed right R-modules. Further, we show that for a wide class of rigs R that includes all rings, these theories are commutants of one another in the full finitary theory of R in the category of sets. We define left R -convex spaces for a preordered ring R as left affine spaces over the positive part \(R_+\) of R. We show that for any firmly archimedean preordered algebra R over the dyadic rationals, the theories of left R-convex spaces and pointed right \(R_+\) -modules are commutants of one another within the full finitary theory of \(R_+\) in the category of sets. Applied to the ring of real numbers \(\mathbb {R}\) , this result shows that the connection between convex spaces and pointed \(\mathbb {R}_+\) -modules that is implicit in the integral representation of probability measures is a perfect ‘duality’ of algebraic theories. PubDate: 2017-05-27 DOI: 10.1007/s10485-017-9496-9

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