Category Of Topological Spaces Research Articles

Models for torsion homotopy types

Given an integern>1 and any setP of positive integers, one can assign to each topological spaceX a homotopy universal mapX(P,n)→X whereX(P,n) is an (n−1)-connected CW-complex whose homotopy groups areP-torsion. We analyze this construction and its properties by means of a suitable closed model category structure on the pointed category of topological spaces.

A classification of types of extensions from TOP to QFTOP

By making use of the concept of the introduced quasi-fuzzy topological space [C.-Y. Zheng, Fuzzy Sets and Systems 14 (1984) 273–280], we introduce the concept of s-extensions. Mainly, we generalize and characterize the concepts of good extensions [R. Lowen, J. Math. Anal. Appl. 56 (1976) 621–633], weak inducement [H.W. Martin, J. Math. Anal. Appl. 78 (1980) 634–639] and s-extensions by introducing some relations between the category of topological spaces and the category of quasi-topological spaces.

On topology and quantum mechanics

The relationship between topology and quantum mechanics is considered in two different ways. We suggest regarding quantum mechanics as a first-order approximation to topology theory on a fixed set. In this spirit, the approach of quantizing the lattice τ(X) of topologies on a set X [C. J. Isham, “Quantum topology and quantization on the lattice of topologies,” Class. Quantum. Grav. 6, 1509–1534 (1989)] is shown to be easily applicable to a quantum mechanical lattice. In the main part of the paper we deal with the question of the quantization of the whole category of topological spaces and continuous injective maps. Ideas of the modern categorical approach to quantization, as known from topological quantum field theory, are invoked in this part. The final paragraphs are devoted to the problem of extending quantization, even to the level of the underlying set theory.

Separated and Connected Maps

Using on the one hand closure operators in the sense of Dikranjan and Giuli and on the other hand left- and right-constant subcategories in the sense of Herrlich, Preus, Arhangel'skii and Wiegandt, we apply two categorical concepts of connectedness and separation/disconnectedness to comma categories in order to introduce these notions for morphisms of a category and to study their factorization behaviour. While at the object level in categories with enough points the first approach exceeds the second considerably, as far as generality is concerned, the two approaches become quite distinct at the morphism level. In fact, left- and right-constant subcategories lead to a straight generalization of Collins' concordant and dissonant maps in the category $$\mathcal{T}op$$ of topological spaces. By contrast, closure operators are neither able to describe these types of maps in $$\mathcal{T}op$$ , nor the more classical monotone and light maps of Eilenberg and Whyburn, although they give all sorts of interesting and closely related types of maps. As a by-product we obtain a negative solution to the ten-year-old problem whether the Giuli–Husek Diagonal Theorem holds true in every decent category, and exhibit a counter-example in the category of topological spaces over the 1-sphere.

Coreflections Versus Regular Epimorphisms in Categories of Topological Spaces

Let \(\mathcal{A}\) be an epireflective subcategory of the category Top of topological spaces which is not contained in the category of indiscrete spaces (e.g. Top, the category of Hausdorff spaces, the category of Tychonoff spaces) and \(\mathcal{B}\) be a coreflective subcategory of \(\mathcal{A}\). In this paper we prove that the coreflector\(\mathcal{A} \to \mathcal{B}\) preserves regular epimorphisms if and only if \(\mathcal{B} = \mathcal{A}\) or \(\mathcal{B}\) is contained in the category of discrete spaces.

Algebraic theories of compact pospaces

Let CmptPoSp denote the category of compact pospaces with continuous monotone maps and let PoSet denote the category of partially ordered sets and monotone maps. In this paper we show that the forgetful functor G: CmptPoSp → PoSet is monadic; that is, G has a left-adjoint and CmptPoSp is isomorphic to the category of algebras PoSet B for the monad B on PoSet induced by the adjunction. This result, which is an asymmetric version of Manes' theorem, shows that the notion of compact pospace is algebraic in a precise sense and provides a useful tool for investigating the category CmptPoSp. As a corollary we obtain the theorem of Simmons and Wyler which says that CmptPoSp is also algebraic over the category of topological spaces and continuous maps. This makes explicit the connection between the Salbany and the prime Wallman compactifications. We also give an explicit construction—as the prime spectrum of the lattice of upper sets—of the Stone-Čech-Nachbin order compactification for a discrete ordered space.

Tensor Products in Categories of Topological Spaces

In this paper symmetric monoidal closed structures on coreflective subcategories of the category of (Hausdorff) topological spaces are studied. We describe all such structures on the category of (Hausdorff) pseudoradial spaces and some of its subcategories and give an example of a coreflective subcategory of the category of Hausdorff topological spaces admitting a proper class of symmetric monoidal closed structures.

Duality beyond sober spaces: Topological spaces and observation frames

We introduce observation frames as an extension of ordinary frames. The aim is to give an abstract representation of a mapping from observable predicates to all predicates of a specific system. A full subcategory of the category of observation frames is shown to be dual to the category of T 0 topological spaces. The notions we use generalize those in the adjunction between frames and topological spaces in the sense that we generalize finite meets to infinite ones. We also give a predicate logic of observation frames with both infinite conjunctions and disjunctions, just like there is a geometric logic for (ordinary) frames with infinite disjunctions but only finite conjunctions. This theory is then applied to two situations: firstly to upper power spaces, and secondly we restrict the adjunction between the categories of topological spaces and of observation frames in order to obtain dualities for various subcategories of T 0 spaces. These involve nonsober spaces.

Extensions of chain functors

Let K be a category of topological spaces and P ⊂ K a subcategory of good spaces; let moreover C K (resp. C p ) be the associated category of chain functors (cf. 1.3.) on K (resp. on P ) and the restriction functor. The main theorem (theorem 1.4.2.) asserts the existence of a kind of right adjoint with the following properties: 1) s C * is c-continuous (cf. 1.2.); 2) if determine isomorphic homology theories (on P ) , then so do (now on the larger category K ); 3) determine isomorphic homology theories on P. The main application of this result consists in an existence proof of a strong homology theory sh* on K agreeing with a prescribed sh* on P Suppose that C ϵ C p determines h * (on P ) then s h * will be the derived homology of s C * Due to properties 1) - 3) of s( ), this s h * is well-defined (in fact uniquely determined by h * and the axioms of a strong homology theory)c-continuous (constituting one of these axioms, which replaces Milnor’s clusteraxiom) and agrees, up to an isomorphism, with h * on ...

Pairings in Categories

We study a categorical generalization of some of the basic results of the homotopy theory which have been obtained in the category of topological spaces with base point. They are some of the results of Gottlieb, Hilton, Hoo, Lim, Varadarajan, Woo and Kim and others. To carry out categorical approach, we define pseudo-product, pseudo-coproduct and homotopy relation in general categories. These concepts enable us to define pairings and copairings in categories. Some of the classical results mentioned above are generalized to the categories with product, coproduct and zero object.

Pairings of Homotopy Sets Over and Under B

AbstractWe define pairings and copairings in the category of topological spaces over and under B. We prove a relation between pairings of homotopy sets induced by a pairing and a copairing of topological spaces over and under B. We obtain some properties of the axes of pairings and the homotopy set of the axes of pairings in the category of topological spaces over and under B. We also prove the dual results.

Isomorphism conjectures in algebraic 𝐾-theory

duff In this paper we are concerned with the four functors Y., Y3 if, *, and *, which map from the category of topological spaces X to the category diff of Q-spectra. The functor 3*( ) (or ?if ( )) maps the space X to the Qspectrum of stable topological (or smooth) pseudoisotopies of X. (The spaces diff dif 3i(X), id9 (X), i > 0, are Hatcher's deloopings of 970(X), % Yd(X) (cf. [29])). The functor J*( ) maps a path-connected space X to the algebraic Ktheoretic S2-spectrum for the integral group ring Z7r1X. (The spaces X(X), i > 0, are the Gersten-Wagoner deloopings of Z%O(X); cf. [27, 52].) 5*(X) is also defined on a nonpath-connected space X to be () (X1), where {Xi: i e I} are the path components of X and iI (Xi) indicates the direct limit of all finite products of the {I (Xi): i E I}. If X is path connected then the functor *-' ( ) maps X to the L -surgery classifying spaces for oriented surgery problems with fundamental group r I1X identified by their fourfold periodicity k 90(X) = k+4j (X) (cf. [35, 39, 44, 47]). In addition, if X is not path connected then we set Y77(X) = tiEI5*fJ'(Xi), where {Xi: i E i} are the path components of X. Results obtained by the authors over the past five years (cf. [15-25] and, in particular, [19; 16, A.1, A.18]), together with many earlier results and conjectures which are reviewed in 1.6.1-1.6.6, suggest that, for any connected CWcomplex X and for 5* () denoting any of the above functors, the Q-spectrum *(X) should be computable in a simple way from the values taken by 9'*() on covering spaces of X, which have very simple fundamental groups. In more detail, we let 9'(X) denote the collection of all subgroups H c ir IX, which are either finite or virtually infinite cyclic. A group H is virtually infinite cyclic if there is a short exact sequence of groups 1 -+ Z -+ H -+ F -+ 1 with F equal to a finite group and Z equal to the infinite cyclic group. For each such group we let XH -+ X denote the covering space projection corresponding to H. Then the Q-spectrum 5? (X) should be computable in a simple way from the Q-spectra {5 (XH): H E 9 (X)}. The purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any X

Pairings and Copairings in the Category of Topological Spaces

We prove a theorem which gives a relation between the pairings of homotopy sets induced by pairings and copairings of topological spaces. We obtain many results on commutativity of elements of homotopy sets as immediate consequences of the theorem. As a further application of the theorem, we prove a generalization of a theorem of Sugawara on the existence of inverse element in the homotopy set when the target space is a Hopf space. We also prove the dual result which is a generalization of a theorem of Hilton, Mislin and Roitberg.

Objects dual to subsemigroups of groups

In this paper we give a description of the duality between closed submonoids of topological groups (Lie groups) and certain cones of continuous (smooth, analytic) functions in a categorial setting. We consider the categories of topological spaces, smooth manifolds and analytic manifolds. A direct approach in the category of smooth manifolds can already be found in [4]. We use the properties of heat kernels on Lie groups to obtain the results in the analytic category. Finally we give a characterization of those Lie wedges which admit strictly positive analytic functions which is a stronger version of results in [3] and [4]. This characterization has various applications in the globality theory of Lie wedge ([3], [4], [7], [9]) and also to ordered homogeneous spaces ([6]) and harmonic analysis on ordered symmetric spaces ([9]).

Topological representation for fuzzy topological spaces

Abstract Let L be a completely distributive complete lattice with universal bounds 0 and 1 such that 0 ≠ 1. Let us denote by X ∗ (L) the of set all prime proper filters of L. To every L-fuzzy topological space (X, τ) one can associate a topological ( X ∗ (L), T ∗ ). One obtains a covariant functor from the category CG (L) into the category of topological spaces. We discuss further how far one can go in this direction following ideas in [2].

Covariant functors in categories of topological spaces

This survey is devoted to the properties of certain concrete covariant functors-normal and almost normal functors-in the category of compacta, as well as the algebraic theory of covariant functors, and the connections between the theory of functors with absolute extensors and manifolds.

Model theory and spectra

Given an extension T of the theory of commutative rings with 1 admitting elimination of quantifiers, we set up a spectrum functor T -Spec from the category of commutative rings with 1 into the category of topological spaces. T -Spec A consists of equivalence classes of homomorphisms of A into models of T . ‘Closed’ formulae - defined as those equivalent to a positive quantifier-free formulae - are used to define a topology on T -Spec A making it into a spectral space. Additional properties, such as the existence of ‘enough prime models’ and ‘closed projections’ are defined snd discussed. If there are enough prime models, we define affine space in a way familiar to algebraic geometers. Affine space enjoys an Artin-Lang property. Finally, we define sheaves of definable sections and continuous definable sections which turn out to be classical sheaves in the classical examples of the Zariski and real spectra. Notions such as continuous, closed, bounded, and having stalks which are local rings are defined in a purely model- theoretic way, and we are able to prove some rather general continuity results about T -Spec as a functor into the category of locally ringed spaces. We pay considerable attention to a new example - the real valuation spectrum - based upon Cherlin and Dickmann's work on the theory of real closed valuation rings. Our set-up allows us to relate continuous definable sections over the real spectrum to definable sections over the real valuation spectrum.

DISCONNECTEDNESS

Abstract There are three different ways to characterize To-spaces in the category of topological spaces. All three methods are canonical, i.e. they can be easily formulated in a general setting, where they, in general, do not coincide. In the following, the characterization of T0-spaces by indiscrete spaces is generalized to an abstract category and investigated.

How Topological are Topological Groups?

Abstract A Characterization of the category of topological groups is provided which does not refer to the category of topological spaces at all, but only to the category of uniform spaces. Similarly, the category of SIN-groups is characterized in a purely uniform way.

Compact nilpotent groups

Herrlich, Salicrup, and Strecker [HSS] have shown that Kuratowski’s Theorem, namely, that a space X is compact if and only if for every space Y, the projection π2X×Y → Y is a closed map, can be interpreted categorically, and hence generalized and applied in a wider settin than the category of topological spaces. The first author, in an earlier paperj [Fl] , applied this categorical interpretation of compactness in categories of R-modules, obtaining a theory of compactness for each torsion theory T. In the case of the category of abelian groups and a hereditary torsion theory T, a group G is T-compact provided G/TG is a T-injective. In this note, the notion of compact is extended to the categories of hypercentral groups, nilpotent groups, and of FC-groups; it is shown that if T π denotes the π-torsion subgroup functor for a set of primes π, then a group G is T π-compact provided G/T πG is π-complete, extending the abelian group result in a natural way.