Authors:Jiří Adámek; Lurdes Sousa Pages: 855 - 872 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 DOI: 10.1007/s10485-018-9530-6 Issue No:Vol. 26, No. 5 (2018)

Authors:B. Banaschewski Pages: 873 - 881 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 DOI: 10.1007/s10485-018-9514-6 Issue No:Vol. 26, No. 5 (2018)

Authors:Ben Berckmoes Pages: 883 - 889 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 DOI: 10.1007/s10485-018-9512-8 Issue No:Vol. 26, No. 5 (2018)

Authors:Maria Manuel Clementino; Jorge Picado; Aleš Pultr Pages: 891 - 906 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 DOI: 10.1007/s10485-018-9516-4 Issue No:Vol. 26, No. 5 (2018)

Authors:Maria Manuel Clementino; Jorge Picado; Aleš Pultr Pages: 907 - 908 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 DOI: 10.1007/s10485-018-9533-3 Issue No:Vol. 26, No. 5 (2018)

Authors:E. Colebunders; M. Sioen; W. Van Den Haute Pages: 909 - 930 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 DOI: 10.1007/s10485-018-9532-4 Issue No:Vol. 26, No. 5 (2018)

Authors:Alan S. Cigoli; Tomas Everaert; Marino Gran Pages: 931 - 942 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 DOI: 10.1007/s10485-018-9515-5 Issue No:Vol. 26, No. 5 (2018)

Authors:Olivier De Deken; Wendy Lowen Pages: 943 - 996 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 DOI: 10.1007/s10485-018-9526-2 Issue No:Vol. 26, No. 5 (2018)

Authors:John Frith; Anneliese Schauerte Pages: 997 - 1013 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 DOI: 10.1007/s10485-018-9521-7 Issue No:Vol. 26, No. 5 (2018)

Authors:P.-A. Jacqmin; S. Mantovani; G. Metere; E. M. Vitale Pages: 1015 - 1039 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 DOI: 10.1007/s10485-018-9529-z Issue No:Vol. 26, No. 5 (2018)

Authors:George Janelidze Pages: 1041 - 1046 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 DOI: 10.1007/s10485-018-9525-3 Issue No:Vol. 26, No. 5 (2018)

Authors:George Janelidze; Ross Street Pages: 1047 - 1064 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 DOI: 10.1007/s10485-018-9524-4 Issue No:Vol. 26, No. 5 (2018)

Authors:George Janelidze; Ross Street Pages: 1065 - 1065 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 DOI: 10.1007/s10485-018-9528-0 Issue No:Vol. 26, No. 5 (2018)

Authors:Wei Li; Dexue Zhang Pages: 1067 - 1093 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 DOI: 10.1007/s10485-018-9527-1 Issue No:Vol. 26, No. 5 (2018)

Authors:Walter Tholen Pages: 1095 - 1111 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 DOI: 10.1007/s10485-018-9513-7 Issue No:Vol. 26, No. 5 (2018)

Authors:Ivan Kobyzev; Ilya Shapiro Abstract: We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter–Drinfeld contramodules for Hopf algebras. PubDate: 2018-10-15 DOI: 10.1007/s10485-018-9544-0

Authors:Fernando Lucatelli Nunes 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: 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:Marcel Erné 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