Category Of Topological Spaces Research Articles

On adjunctions between Lim, SL- Top, and SL- Lim

Consider ( L , * , 1 ) be a commutative, strictly two-sided quantale with the underlying lattice L being meet-continuous. Two adjunctions, one is between limit spaces and stratified L-limit spaces and the other is between stratified L-limit spaces and stratified L-topological spaces, are established. The first adjunction can be viewed as an extension of Lowen's adjunction between the category of topological spaces and stratified [0,1]-topological spaces. The second is an extension of an adjunction between limit spaces and (stratified) L-topological spaces established in U. Höhle and T. Kubiak (Höhle–Kubiak, Semigroup Forum (2007)).

Topological objects in the category EQU

We give a criterion for equivalence between equilogical and topological spaces. This enables us to prove that some interesting categories of topological spaces are complete subcartesian closed subcategories of equilogical spaces.

From quantale algebroids to topological spaces: Fixed- and variable-basis approaches

Using the category of quantale algebroids the paper considers a generalization of the classical Papert–Papert–Isbell adjunction between the categories of topological spaces and locales to partial algebraic structures. It also provides a single framework in which to treat the concepts of quasi, standard and stratified fuzzy topology.

The essential cover and the absolute cover of a schematic space

A theorem of Gleason states that every compact space admits a projective cover. More generally, in the category of topological spaces with continuous maps, covers exist with respect to the full subcategory of extremally disconnected spaces. Such a cover o

Sobriety and spatiality in varieties of algebras

The paper considers a generalization of the classical Papert–Papert–Isbell adjunction between the categories of topological spaces and locales to an arbitrary variety of algebras and illustrates the obtained results by the category of algebras over a given unital commutative quantale.

Categorical Properties of Intuitionistic Topological Spaces.

We obtain some characterizations of continuous, open and closed functions in intuitionistic topological spaces. Moreover we reveal that the category of topological spaces is a bireflective full subcategory of the category of intuitionistic topological spaces.

Categories of Topological Spaces and Scattered Theories

We offer a topological treatment of scattered theories intended to help to explain the parallelism between, on the one hand, the theorems provable using Descriptive Set Theory by analysis of the space of countable models and, on the other, those provable by studying a tree of theories in a hierarchy of fragments of infinintary logic. We state some theorems which are, we hope, a step on the road to fully understanding counterexamples to Vaught's Conjecture. This framework is in the early stages of development, and one area for future exploration is the possibility of extending it to a setting in which the spaces of types of a theory are uncountable.

The Second Obstruction Functor

Abstract A functor responsible for second obstruction problems is defined and investigated on the category of topological spaces. In terms of this functor we formulate and prove a classification theorem for maps, which is the reformulation of all known classification theorems for the second obstruction.

Effective codescent morphisms, amalgamations and factorization systems

The paper deals with the question whether it is sufficient, when investigating the problem of the effectiveness of a descent morphism, to restrict the consideration only to the descent data ( C , γ , ξ ) , where γ lies in a certain morphism class. The notion of a factorization system and the dual to the amalgamation property in the sense of Kiss, Marki, Pröhle and Tholen play the key role in our discussion. It is shown that a category C inherits from a category X the property that all descent morphisms are effective if either X is regular and C is a full coreflective, closed under pullbacks of certain epimorphisms, subcategory of X or C is regular, X has coequalizers and there exists a topological functor C → X . This implies that in the category of topological spaces, all regular monomorphisms are effective codescent morphisms (the result of Mantovani). The same is shown to be valid also for the categories of compact Hausdorff topological spaces, normal topological spaces, Banach spaces, (quasi-)uniform spaces, and (quasi-)proximity spaces. Moreover, the effectiveness of all codescent morphisms is established for the categories of Hausdorff topological spaces and (complete) metric spaces. The internal characterization of such morphisms p : B → E is given for the category of Hausdorff topological spaces, in the case of compact B and regular E .

Universality of Coproducts in Categories of Lax Algebras

Categories of lax \((T,V\,)\)-algebras are shown to have pullback-stable coproducts if \(T\) preserves inverse images. The general result not only gives a common proof of this property in many topological categories but also shows that important topological categories, like the category of uniform spaces, are not presentable as a category of lax \((T,V\,)\)-algebras, with \(T\) preserving inverse images. Moreover, we show that any such category of \((T,V\,)\)-algebras has a concrete, coproduct preserving functor into the category of topological spaces.

Probabilistic Observations and Valuations: (Extended Abstract)

We give a universal property for an “abstract probabilistic powerdomain” based on an analysis of observable properties of probabilistic computation. The universal property determines an abstract notion of integration satisfying the usual equational laws. In the category of topological spaces, the abstract probabilsitic powerdomain is given explicitly by the space of continuous probability valuations with weak topology. Thus our abstract notion of integration coincides with the usual integration for probability valuations. We end by discussing how our approach might adapt to provide “abstract effect spaces” for other computational effects.

Completions, comonoids, and topological spaces

We analyse the category-theoretical structures involved with the notion of continuity within the framework of formal topology. We compare the category of basic pairs to other categories of “spaces” by means of canonically determined functors and show how the definition of continuity is determined in a certain, canonical sense. Finally, we prove a standard adjunction between the (co)algebraic approach to spaces and the category of topological spaces.

Postnikov “Invariants” in 2004

Abstract The very nature of the so-called Postnikov invariants is carefully studied. Two functors, precisely defined, explain the exact nature of the connection between the category of topological spaces and the category of Postnikov towers. On one hand, these functors are in particular effective and lead to concrete machine computations through the general machine program Kenzo. On the other hand, the Postnikov “invariants” will be actual invariants only when an arithmetical decision problem – currently open – will be solved; it is even possible this problem is undecidable.

Cartesian Closed Topological Categories and Tensor Products

The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the Cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X.

Comparing Cartesian closed categories of (core) compactly generated spaces

It is well known that, although the category of topological spaces is not Cartesian closed, it possesses many Cartesian closed full subcategories, e.g.: (i) compactly generated Hausdorff spaces; (ii) quotients of locally compact Hausdorff spaces, which form a larger category; (iii) quotients of locally compact spaces without separation axiom, which form an even larger one; (iv) quotients of core compact spaces, which is at least as large as the previous; (v) sequential spaces, which are strictly included in (ii); and (vi) quotients of countably based spaces, which are strictly included in the category (v). We give a simple and uniform proof of Cartesian closedness for many categories of topological spaces, including (ii)–(v), and implicitly (i), and we also give a self-contained proof that (vi) is Cartesian closed. Our main aim, however, is to compare the categories (i)–(vi), and others like them. When restricted to Hausdorff spaces, (ii)–(iv) collapse to (i), and most non-Hausdorff spaces of interest, such as those which occur in domain theory, are already in (ii). Regarding the Cartesian closed structure, finite products coincide in (i)–(vi). Function spaces are characterized as coreflections of both the Isbell and natural topologies. In general, the function spaces differ between the categories, but those of (vi) coincide with those in any of the larger categories (ii)–(v). Finally, the topologies of the spaces in the categories (i)–(iv) are analyzed in terms of Lawson duality.

SEVERAL CONSTRUCTIONS OF EXACT SEQUENCES

The article presents a construction of a 5-term relative exact sequence in purely categorical terms and the Mayer–Vietoris sequence for weak ∞-groupoids of R. Street. In the first half of the paper, it is shown that several well-known exact sequences can be obtained, using the Z-diagram described at the beginning of the article. There are number of categories that are close to the category of topological spaces in the sense of homotopy theory. The objects of such a category can be viewed as weak ∞-groupoids in a very intuitive way. Therefore, the language of weak ∞-groupoids seems to be very convenient to make explicit constructions in categories of this type. The article presents a combinatorial construction of the path space of a weak ∞-groupoid and applies the machinery of Sec. 1 to obtain the long exact sequence of a homotopy fiber. The construction is quite natural and gives information on the structure of the relative terms. In the last section, the Mayer–Vietoris sequence of a fiber square of weak ∞-groupoids is obtained under some conditions. Of course, this construction makes sense for topological spaces or for any category mentioned above. But the statement of the conditions is more natural in terms of weak ∞-groupoids. This sequence generalizes the sequence of fibration, and in this case the conditions given in the article are equivalent to the homotopy lifting property. Bibliography: 6 titles.

On the Relationship Between Several Basic Categories in Fuzzy Topology

In this paper, a process, which associates with every fibre-small topo- logical category A and every completely distributive lattice L another topological category called the co-tower extension of A, is introduced to investigate the relationship between several basic categories in fuzzy topology. It is proved that the category of stratified Šostak L-fuzzy topological spaces is the co-tower extension of the category of stratified Chang-Goguen spaces; and the category of L-fuzzifying topological spaces (which is isomorphic the category of fuzzy neighborhood spaces when L = [0, 1]) is the co-tower extension of the category of topological spaces. Thus, in some sense, we can say that properties of the category of stratified Šostak L-fuzzy topological spaces and L-fuzzifying topological spaces are determined by the categories of stratified Chang-Goguen spaces, topological spaces and the lattice L. Therefore it can be said that in fuzzy topology the category of (stratified) Chang-Goguen topological spaces is more basic and more important than the other categories.

DW Complexes and Their Underlying Topological Spaces

The naive concept of a DW complex is that of a differential space that can be built up from cells and whose differential structure is defined in terms of differential structures on Euclidean unit closed balls. This concept stems from an analogue in the category of topological spaces: the so-called CW complex. The paper goes as far as investigating the underlying topological space of a DW complex.

Heredity and Coreflective Subcategories of the Category of Topological Spaces

Every hereditary coreflective subcategory of Top containing the category of finitely-generated spaces is shown to be generated by a class of spaces having a unique accumulation point. It is also shown that the coreflective hull of a union of two hereditary coreflective subcategories of Top need not be hereditary so that a coreflective subcategory of Top need not have a hereditary coreflective kernel.

The monad induced by the hom‐functor in the category of topological spaces and its associated Eilenberg‐Moore algebras

We discuss the monad associated with the topology of pointwise convergence. We also study examples of the Eilenberg‐Moore algebras for this monad.