Abstract: In the original publication of the article, Eq. (3.24) was published incorrectly. The corrected equation is given in this correction article. The original article has been corrected. PubDate: 2018-10-01

Abstract: In the original publication of the article, the formulation of the c-subfitness condition (c-sfit) in Subsection 5.2 is inaccurate, with effect in Theorem 5.3. PubDate: 2018-10-01

Abstract: Abstract We describe an alternative way of constructing some of the monads, recently introduced by E. Colebunders, R. Lowen, and W. Rosiers for the purposes of categorical topology. PubDate: 2018-10-01

Abstract: Abstract Sublocales of a locale (frame, generalized space) can be equivalently represented by frame congruences. In this paper we discuss, a.o., the sublocales corresponding to complete congruences, that is, to frame congruences which are closed under arbitrary meets, and present a “geometric” condition for a sublocale to be complete. To this end we make use of a certain closure operator on the coframe of sublocales that allows not only to formulate the condition but also to analyze certain weak separation properties akin to subfitness or \(T_1\) . Trivially, every open sublocale is complete. We specify a very wide class of frames, containing all the subfit ones, where there are no others. In consequence, e.g., in this class of frames, complete homomorphisms are automatically Heyting. PubDate: 2018-10-01

Abstract: Abstract For a functor F whose codomain is a cocomplete, cowellpowered category \(\mathcal {K}\) with a generator S we prove that a codensity monad exists iff for every object s in S all natural transformations from \(\mathcal {K}(X, F-)\) to \(\mathcal {K}(s, F-)\) form a set. Moreover, the codensity monad has an explicit description using the above natural transformations. Concrete examples are presented, e.g., the codensity monad of the power-set functor \(\mathcal {P}\) assigns to every set X the set of all nonexpanding endofunctions of \(\mathcal {P}X\) . Dually, a set-valued functor F is proved to have a density comonad iff all natural transformations from \(X^F\) to \(2^F\) form a set. Moreover, that comonad assigns to X the set of all those transformations. For preimages-preserving endofunctors F of \({\mathsf {Set}}\) we prove that F has a density comonad iff F is accessible. PubDate: 2018-10-01

Abstract: Abstract For the \(\ell \) -ring F(L) introduced in Karimi Feizabadi et al. (Categ Gen Algebr Struct Appl 5:85–102, 2016) and then shown to have an embedding into the familiar \(\ell \) -ring \({\mathfrak R}L\) of all real-valued continuous function on a frame L, the resulting image \({\mathfrak S}L\) in \({\mathfrak R}L\) is characterized here by internal properties within \({\mathfrak R}L\) . Further, a number of results concerning the \({\mathfrak S}L\) are obtained on the basis of this characterization. PubDate: 2018-10-01

Abstract: Abstract The congruence lattice of a frame has long been an object of considerable interest, not least because it turns out to be a frame itself. Perhaps more surprisingly congruence lattices of, for instance, \(\sigma \) -frames, \(\kappa \) -frames and some partial frames also turn out to be frames. The situation for congruences of a meet-semilattice is notably different. In this paper we analyze the meet-semilattice congruence lattices of arbitrary frames and compare them with the corresponding lattices of frame congruences. In the course of this, we provide a structure theorem as well as many examples and counter-examples. PubDate: 2018-10-01

Abstract: Abstract We establish an approach theoretic version of Anscombe’s theorem, which we apply to justify the use of confidence intervals based on the sample mean after a group sequential trial. PubDate: 2018-10-01

Abstract: Abstract Given an exact category \({\mathcal {C}}\) , it is well known that the connected component reflector \( \pi _0 :\mathsf {Gpd}(\mathcal {C}) \rightarrow \mathcal {C}\) from the category \(\mathsf {Gpd}(\mathcal {C})\) of internal groupoids in \(\mathcal {C}\) to the base category \(\mathcal {C}\) is semi-left-exact. In this article we investigate the existence of a monotone-light factorization system associated with this reflector. We show that, in general, there is no monotone-light factorization system \((\mathcal {E}',\mathcal {M}^*)\) in \(\mathsf {Gpd}\) ( \(\mathcal {C}\) ), where \(\mathcal {M}^*\) is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where \(\mathcal {C}\) is an exact Mal’tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in \(\mathsf {Gpd}\) ( \(\mathcal {C}\) ) is the relative monotone-light factorization system (in the sense of Chikhladze) in the category \(\mathsf {Gpd}\) ( \(\mathcal {C}\) ) corresponding to the connected component reflector, where \(\mathcal {E}'\) is the class of final functors and \( \mathcal {M}^*\) the class of regular epimorphic discrete fibrations. PubDate: 2018-10-01

Abstract: Abstract When replacing the non-negative real numbers with their addition by a commutative quantale \(\mathsf{V}\) , under a metric lens one may then view small \(\mathsf{V}\) -categories as sets that come with a \(\mathsf{V}\) -valued distance function. The ensuing category \(\mathsf{V}\text {-}\mathbf{Cat}\) is well known to be a concrete topological category that is symmetric monoidal closed. In this paper we show which concrete symmetric monoidal-closed topological categories may be fully and bireflectively embedded into \(\mathsf{V}\text {-}\mathbf{Cat}\) , for some \(\mathsf{V}\) . PubDate: 2018-10-01

Abstract: Abstract For approach spaces normality has been studied from different angles. One way of dealing with it is by focussing on separation by realvalued contractive maps or, equivalently, on Katětov–Tong’s insertion. We call this notion approach normality. Another point of view is using the isomorphism between the category \(\textsf {App}\) of approach spaces and contractions and the category of lax algebras for the ultrafilter monad and the quantale \(\textsf {P}_{\!\!{_+}}\) and applying the monoidal definition of normality. We call this notion monoidal normality. Although both normality properties coincide for topological approach spaces, a comparison of both notions for \(\textsf {App}\) is an open question. In this paper we present a partial solution to this problem. We show that in \(\textsf {App}\) approach normality implies monoidal normality and that both notions coincide on the subcategory of quasimetric approach spaces. Moreover we investigate the relation between approach normality and regularity. Among other things we prove that approach spaces that are approach normal and regular are uniform. PubDate: 2018-10-01

Abstract: Abstract We develop the basic theory of curved \(A_{\infty }\) -categories ( \(cA_{\infty }\) -categories) in a filtered setting, encompassing the frameworks of Fukaya categories (Fukaya et al. in Part I, AMS/IP studies in advanced mathematics, vol 46, American Mathematical Society, Providence, RI, 2009) and weakly curved \(A_{\infty }\) -categories in the sense of Positselski (Weakly curved \(A_\infty \) algebras over a topological local ring, 2012. arxiv:1202.2697v3). Between two \(cA_{\infty }\) -categories \(\mathfrak {a}\) and \(\mathfrak {b}\) , we introduce a \(cA_{\infty }\) -category \(\mathsf {qFun}(\mathfrak {a}, \mathfrak {b})\) of so-called \(qA_{\infty }\) -functors in which the uncurved objects are precisely the \(cA_{\infty }\) -functors from \(\mathfrak {a}\) to \(\mathfrak {b}\) . The more general \(qA_{\infty }\) -functors allow us to consider representable modules, a feature which is lost if one restricts attention to \(cA_{\infty }\) -functors. We formulate a version of the Yoneda Lemma which shows every \(cA_{\infty }\) -category to be homotopy equivalent to a curved dg category, in analogy with the uncurved situation. We also present a curved version of the bar-cobar adjunction. PubDate: 2018-10-01

Abstract: Abstract We make various observations on infinitary addition in the context of the series monoids introduced in our previous paper on real sets. In particular, we explore additional conditions on such monoids suggested by Tarski’s Arithmetic of Cardinal Algebras, and present a monad-theoretic construction that generalizes our construction of paradoxical real numbers. PubDate: 2018-10-01

Abstract: Abstract We characterize fibrations and \(*\) -fibrations in the 2-category of internal groupoids in terms of the comparison functor from certain pullbacks to the corresponding strong homotopy pullbacks. As an application, we deduce the internal version of the Brown exact sequence for \(*\) -fibrations from the internal version of the Gabriel–Zisman exact sequence. We also analyse fibrations and \(*\) -fibrations in the category of arrows and study when the normalization functor preserves and reflects them. This analysis allows us to give a characterization of protomodular categories using strong homotopy kernels and a generalization of the Snake Lemma. PubDate: 2018-10-01

Abstract: Abstract The notion of Scott distance between points and subsets in a metric space, a metric analogy of the Scott topology on an ordered set, is introduced, making a metric space into an approach space. Basic properties of Scott distance are investigated, including its topological coreflection and its relation to injective \(T_0\) approach spaces. It is proved that the topological coreflection of the Scott distance is sandwiched between the d-Scott topology and the generalized Scott topology; and that every injective \(T_0\) approach space is a cocomplete and continuous metric space equipped with its Scott distance. PubDate: 2018-10-01

Authors:Fernando Lucatelli Nunes Abstract: Abstract Given a pseudomonad \(\mathcal {T}\) , we prove that a lax \(\mathcal {T}\) -morphism between pseudoalgebras is a \(\mathcal {T}\) -pseudomorphism if and only if there is a suitable (possibly non-canonical) invertible \(\mathcal {T}\) -transformation. This result encompasses several results on non-canonical isomorphisms, including Lack’s result on normal monoidal functors between braided monoidal categories, since it is applicable in any 2-category of pseudoalgebras, such as the 2-categories of monoidal categories, cocomplete categories, bicategories, pseudofunctors and so on. PubDate: 2018-09-15 DOI: 10.1007/s10485-018-9541-3

Authors:Jorge Picado; Aleš Pultr Abstract: Abstract We study uniformities and quasi-uniformities (uniformities without the symmetry axiom) in the common language of entourages. The techniques developed allow for a general theory in which uniformities are the symmetric part. In particular, we have a natural notion of Cauchy map independent of symmetry and a very simple general completion procedure (perhaps more transparent and simpler than the usual symmetric one). PubDate: 2018-09-15 DOI: 10.1007/s10485-018-9542-2

Authors:Jaime Castro Pérez; Mauricio Medina Bárcenas; José Ríos Montes; Angel Zaldívar Corichi Abstract: Abstract We are concerned with the boolean or more generally with the complemented properties of idioms (complete upper-continuous modular lattices). Simmons (Cantor–Bendixson, socle, and atomicity. http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/002-Atom.pdf, 2014) introduces a device which captures in some informal speaking how far the idiom is from being complemented, this device is the Cantor-Bendixson derivative. There exists another device that captures some boolean properties, the so-called Boyle-derivative, this derivative is an operator on the assembly (the frame of nuclei) of the idiom. The Boyle-derivative has its origins in module theory. In this investigation we produce an idiomatic analysis of the boolean properties of any idiom using the Boyle-derivative and we give conditions on a nucleus j such that [j, tp] is a complete boolean algebra. We also explore some properties of nuclei j such that \(A_{j}\) is a complemented idiom. PubDate: 2018-09-15 DOI: 10.1007/s10485-018-9543-1

Authors:Marcel Erné Abstract: Abstract A subset of a topological space is hypercompact if its saturation (the intersection of its neighborhoods) is generated by a finite set. Locally hypercompact spaces are defined by the existence of hypercompact neighborhood bases at each point. We exhibit many useful properties of such spaces, often based on Rudin’s Lemma, which is equivalent to the Ultrafilter Principle and ensures that the Scott spaces of quasicontinuous domains are exactly the locally hypercompact sober spaces. We characterize their patch spaces (the Lawson spaces) as hyperconvex and hyperregular pospaces in which every monotone net has a supremum to which it converges. Moreover, we find extensions to the non-sober case by replacing suprema with cuts, and we provide topological generalizations of known facts for quasicontinuous posets. Similar results are obtained for hypercompactly based spaces and quasialgebraic posets. Furthermore, locally hypercompact spaces are described by certain relations between finite sets and points, providing a quasiuniform approach to such spaces. Our results lead to diverse old and new equivalences and dualities for categories of locally hypercompact spaces or quasicontinuous posets. PubDate: 2018-08-09 DOI: 10.1007/s10485-018-9536-0