Category Of Topological Spaces Research Articles

Universal Finite Functorial Semi-norms

Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-semi-norm. However, the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.

Localic separation and the duality between closedness and fittedness

There are a number of localic separation axioms which are roughly analogous to the T1-axiom from classical topology. For instance, besides the well-known subfitness and fitness, there are also Rosický–Šmarda's T1-locales, totally unordered locales and, more categorically, the recently introduced F-separated locales (i.e., those with a fitted diagonal) — a property strictly weaker than fitness.It has recently been shown that the strong Hausdorff property and F-separatedness are in a certain sense dual to each other. In this paper, we provide further instances of this duality — e.g., we introduce a new first-order separation property which is to F-separatedness as the Johnstone–Sun-shu-Hao–Paseka–Šmarda conservative Hausdorff axiom is to the strong Hausdorff property, and which can be of independent interest. Using this, we tie up the loose ends of the theory by establishing all the possible implications between these properties and other T1-type axioms occurring in the literature. In particular, we show that the strong Hausdorff property does not imply F-separatedness, a question which remained open and shows a remarkable difference with its counterpart in the category of topological spaces.

OVERVIEW ON MODELS IN HOMOTOPICAL ALGEBRA

A covariant functor ∆ → A is referred to as a model object of A, generating a subject matter in A analogous to algebraic topology when A is the category of topological spaces; this work delineates the contexts in which these concepts are developed and outlines the key contributions made by the author concerning model objects.

Separation, connectedness, and disconnectedness

The aim of this paper is to introduce the notions of hereditarily disconnected and totally disconnected objects in a topological category and examine the relationship as well as interrelationships between them. Moreover, we characterize each of $T_{2}$, connected, hereditarily disconnected, and totally disconnected objects in some topological categories and compare our results with the ones in the category of topological spaces.

Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category.

Coalgebraic Geometric Logic: Basic Theory

Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category.

On the globalization of geometric partial (co)modules in the categories of topological spaces and algebras

We study the globalization of partial actions on sets and topological spaces and of partial coactions on algebras by applying the general theory of globalization for geometric partial comodules, as previously developed by the authors. We show that this approach does not only allow to recover all known results in these settings, but it allows to treat new cases of interest, too.

WHITNEY APPROXIMATION FOR SMOOTH CW COMPLEX

We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists a partition of unity subordinate to the open covering.

Sober spaces

The goal of this paper is to introduce various forms of sober objects in a topological category and investigate the relationships among these various forms. Moreover, we characterize quasi-sober objects, $T_{0}$ objects and each of various forms of sober objects in some topological categories and give some invariance properties of them. Finally, we compare our results with some well-known results in the category of topological spaces.

Probability, valuations, hyperspace: Three monads on top and the support as a morphism

AbstractWe consider three monads on $\mathsf{Top}$ , the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second one is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them and allows us to prove that the operation of taking the support of a continuous valuation is a morphism of monads $V \to H$ . In particular, this implies that every H-algebra (topological complete semilattice) is also a V-algebra. We show that V can be restricted to a submonad of $\tau$ -smooth probability measures on $\mathsf{Top}$ . By composing these morphisms of monads, we obtain that taking the supports of $\tau$ -smooth probability measures is also a morphism of monads.

Some Properties of the Covariant Functor Set of Exponential Type

In this paper, it is shown that the sets of all non-empty subsets Set (x) of a topological space X with exponential topology is a covariant functor in the category of -topological spaces and their continuous mappings into itself. It is shown that the functor Set is a covariant functor in the category of topological spaces and continuous mappings into itself, a pseudometric in the space Set (x) is defined, and compact, connected, finite, and countable subspaces of Set (x) are distinguished. It also shows various kinds of connectivity, soft, locally soft, and n - soft mappings in Set (x). One interesting example is given for the TOPY category. It is proved that the functor Set maps open mappings to open, contractible and locally contractible spaces and into contractible and locally contractible spaces.

Mappings on intuitionistic fuzzy topology of soft sets

The present study is devoted to describe the concepts of continuous mapping, open mapping and closed mapping by using soft points on intuitionistic fuzzy topological spaces. Along, continuous mapping, open mapping and closed mapping on intuitionistic fuzzy topological spaces and their characterizations are also introduced. At the end, some of the crucial properties of the proposed concepts are investigated. Taking advantage of intuitionistic fuzzy topology, we obtain the family of soft topologies and normal topologies. It is clear that the category of intuitionistic fuzzy topological spaces is an extension both the category of soft topological spaces and the category of topological spaces.

Models for subhomogeneous $C^*$-algebras

A new category of topological spaces with additional structures, called m-towers, is introduced. It is shown that there is a covariant functor which establishes a one-to-one correspondences between unital (resp. arbitrary) subhomogeneous C*-algebras and proper (resp. proper pointed) m-towers of finite height, and between all *-homomorphisms between two such algebras and morphisms between m-towers corresponding to these algebras.

Raney Algebras and Duality for $$T_0$$-Spaces

In this note we adapt the treatment of topological spaces via Kuratowski closure and interior operators on powersets to the setting of $$T_0$$ -spaces. A Raney lattice is a complete completely distributive lattice that is generated by its completely join prime elements. A Raney algebra is a Raney lattice with an interior operator whose fixpoints completely generate the lattice. It is shown that there is a dual adjunction between the category of topological spaces and the category of Raney algebras that restricts to a dual equivalence between $$T_0$$ -spaces and Raney algebras. The underlying idea is to take the lattice of upsets of the specialization order with the restriction of the interior operator of a space as the Raney algebra associated to a topological space. Further properties of topological spaces are explored in the dual setting of Raney algebras. Spaces that are $$T_1$$ correspond to Raney algebras whose underlying lattices are Boolean, and Alexandroff $$T_0$$ -spaces correspond to Raney algebras whose interior operator is the identity. Algebraic description of sober spaces results in algebraic considerations that lead to a generalization of sober that lies strictly between $$T_0$$ and sober.

Another note on effective descent morphisms of topological spaces and relational algebras

We make three independent observations on characterizing effective descent morphisms in the category of topological spaces. The first of them proposes a new modification of known characterizations of effective descent morphisms of general spaces, while the other two are devoted to locally finite and Hausdorff spaces, respectively. The Hausdorff case is considered, as far as we could, at the more general level of relational algebras in the sense of M. Barr.

Manifold calculus adapted for simplicial complexes

We develop a generalization of manifold calculus in the sense of Goodwillie-Weiss where the manifold is replaced by a simplicial complex. We consider functors from the category of open subsets of a fixed simplical complex into the category of topological spaces and prove an analogue of the approximation theorem. Namely, under certain conditions such a functor can be approximated by a tower of (appropriately adapted) polynomial functors.

The strength of prime separation, sobriety, and compactness theorems

We investigate in ZF set theory without choice principles a general lattice-theoretical prime separation lemma and compare it with diverse statements about variants of the sobriety concept for topological spaces. Some of these properties coincide in the presence of choice principles but differ in their absence. UP, the Ultrafilter Principle (or, equivalently, the Prime Ideal Theorem) is not only equivalent to the Separation Lemma, but also necessary and sufficient for the desired coincidences. Furthermore, we prove the equivalence of UP to several statements about filtered systems of compact sets, among them the Hofmann–Mislove Theorems, several compact intersection theorems, and an irreducible transversal theorem. Moreover, many fundamental dualities between certain categories of topological spaces and categories of ordered structures turn out to be equivalent to UP. But we also give choice-free proofs for such dualities, amending slightly the involved definitions.

Spaces of gestures are function spaces

From the point of view of musical performance, gestures are the movements of the body of the performer when playing an instrument. This vague idea can be modeled mathematically, by mixing category theory and topology, giving rise to the definition of a topological gesture with a given skeleton and body in a topological space. The skeleton represents the abstract configuration of the body's limbs and the topological space is a generalization of the three-dimensional space where the body's movements are usually modeled. The collection of all gestures with the same skeleton and body in a fixed space has a canonical topology, yielding a space of gestures. This article intends to show that the space of gestures is homeomorphic to the function space , endowed with the compact-open topology. The topology of this space is the most natural choice for a space of functions, in the sense that it is related to the universal property of exponentials in the category of topological spaces. In particular, when the skeleton has a suitable property of finiteness, we show that the function space becomes a true exponential.

An order-theoretic perspective on categorial closure operators

This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco’s allow one to see more easily the connections between standard conditions on general cco’s, and furthermore, we show that these connections for cco’s can be even deduced from the corresponding ones for bco’s, when considering cco’s relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco’s via bco’s is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco’s, independent results on cco’s are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco’s also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.

The Quillen model category of topological spaces

We give a complete and careful proof of Quillen’s theorem on the existence of the standard model category structure on the category of topological spaces. We do not assume any familiarity with model categories.