Strong pro-fibrations and ANR objects in pro-categories

The notions of pro-fibration and approximate pro-fibration for morphisms in the pro-category pro-Top of topological spaces were introduced by S. Mardeši? and T.B. Rushing. In this paper we introduce the notion of strong pro-fibration, which is a pro-fibration with some additional property, and the notion of ANR object in pro-Top, which is approximately an ANR-system, and we consider the full subcategory ANR of pro-Top whose objects are ANR objects. We prove that the category ANR satisfies most of the axioms for fibration category in the sense of H.J. Baues if fibrations are strong pro-fibrations and weak equivalences are morphisms inducing isomorphisms in the pro-homotopy category pro-H(Top) of topological spaces. We give various applications. First of all, we prove that every shape morphism is represented by a strong pro-fibration. Secondly, the fibre of a strong pro-fibration is well defined in the category ANR, and we obtain an isomorphism between the pro-homotopy groups of the base and total systems of a strong pro-fibration, and hence obtain the pro-homotopy sequence of a strong pro-fibration. Finally, we also show that there is a homotopy decomposition in the category ANR.     

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category S-Top of stratified spaces, that are topological spaces X endowed with a partition F and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,F) of S-Top together with a class A of subsets of X; they are similar to invariants introduced by M. Clapp and D. Puppe.If (X,F)∈S-Top, we define a transverse subset as a subspace A of X such that the intersection S∩A is at most countable for any S∈F. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a C¹-foliation, the three previous definitions, with A the class of transverse subsets, coincide with the tangential category and are homotopical invariants.     

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ _T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.An important special case is the category Sep_Top/ _T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ _T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.     

More on upper bicompletion-true functorial quasi-uniformities

Let T:QU₀→Top₀ denote the usual forgetful functor from the category of quasi-uniform T₀-spaces to that of the topological T₀-spaces. We regard the bicompletion reflector as a (pointed) endofunctor K:QU₀→QU₀. For any section F:Top₀→QU₀ of T we consider the (pointed) endofunctor R=TKF:Top₀→Top₀. The T-section F is called upper bicompletion-true (briefly, upper K-true) if the quasi-uniform space KFX is finer than FRX for every X in Top₀. An important known characterisation is that F is upper K-true iff the canonical embedding X→RX is an epimorphism in Top₀ for every X in Top₀. We show that this result admits a simple, purely categorical formulation and proof, independent of the setting of quasi-uniform and topological spaces. We thus mention a few other settings where the result is applicable. Returning then to the setting T:QU₀→Top₀, we prove: Any T-section F is upper K-true iff for all X the bitopology of KFX equals that of FRX; and iff the join topology of KFX equals the strong topology (also called the b- or Skula topology) of RX.     

Normal subobjects of topological groups and of topological semi-Abelian algebras

As any category Gp(E) of internal groups in a given category E, the category Gp(Top) of topological groups possesses the strong algebraic property of protomodularity which carries intrinsic notions of normal subobject and of centrality. Here we explicit and investigate these intrinsic notions in the category Gp(Top). We extend these results to any category TopT of topological semi-Abelian algebras.     

Fibrewise extensions, shanin compactification and extensions of fibrewise maps

We introduce an alternative definition of fibrewise uniformity and discuss consequences deduced from new axioms. By modifying James’ definition of fibrewise uniform structure, which is a slightly strengthened one, we define a new fibrewise uniformity which is symmetric in global and realizes 1-1 correspondence between fibrewise entourage uniformities and fibrewise covering uniformities. Moreover, we obtain a characterization of the fibrewise completion of fibrewise generalized uniform space as a fibrewise extension of a fibrewise space. As an application of the fibrewise completion theory, we show that there exists a fibrewise Shanin compactification of a fibrewise space. Finally, we study extendability of fibrewise maps from dense subspaces. That is, for a fibrewise space X, A ? X dense in X and a fibrewise continuous map f: A → Y, when can f be extended to the whole space X? Many characterization theorems of extendable fibrewise continuous maps are given.     

Fibrewise injectivity and Kock–Zöberlein monads

Using Escardó’s characterization of injectivity via Kock–Zöberlein monads, we introduce suitable monads in comma categories of topological spaces that yield characterizations of fibrewise injectivity in topological T0-spaces, with respect to the class of embeddings, and of dense, of flat and of completely flat embeddings. Characterizations, in the category of topological spaces, of injective maps with respect to the same classes of embeddings follow easily from the results obtained for T0-spaces. Moreover, it is shown that, together with the corresponding embeddings, injective continuous maps form a weak factorization system in the category of topological (T0-)spaces and continuous maps.     

Precosheaves of pro-sets and abelian pro-groups are smooth

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

2-Categorical aspects of strong shape

For every pair (C,K) of groupoid enriched categories, we define two related categories S(C,K) and SS(C,K). In the case (C,K)=(TOP,ANR) they are isomorphic, respectively, to the classical shape category Sh(TOP) of Mardeši? and Segal and to the strong shape category of topological spaces SSh(1)(TOP) of height 1, defined by Lisicá and Mardeši?. Moreover, for (C,K)=(CM,ANR), where CM is the category of compact metric spaces, there is an isomorphism SSh(CM)≅SS(CM,ANR). A new characterization of topological strong shape equivalences is also given.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

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. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Exponential homeomorphisms in the category of topological spaces with base point

Topologies on base point preserving function spaces are studied making use of the Brown-Booth-Tillotson C-smash product and the topologies on function spaces defined by the classes C of exponentiable spaces. Some conditions on the classes C are obtained for exponential bijections and exponential homeomorphisms in the category of topological spaces with base point. Conditions for exponential homeomorphisms are obtained for CW-complexes whose cellular structure implies precise results.     

Conway’s Question: The Chase for Completeness

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–?ech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw _C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Ra?kov completion ${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{G}}$ . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.     

Uniform universal covers of uniform spaces

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. The uniform universal cover of a coverable space is a kind of generalized cover with universal and lifting properties in the category of uniform spaces and uniformly continuous mappings. Associated with the uniform universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces.     

L-fuzzy Q-convergence structures

In this paper, we construct a topological category of pretopological L-fuzzy Q-convergence spaces, which contains the category of topological L-fuzzy Q-convergence spaces as a bireflective full subcategory. Considering the connections with L-fuzzy topology, it is proved that the category of topological L-fuzzy Q-convergence spaces is isomorphic to the category of topological L-fuzzy quasi-coincident neighborhood spaces, and the latter is isomorphic to the category of L-fuzzy topological spaces. Moreover, we find that our pretopological L-fuzzy Q-convergence spaces can be characterized as a kind of L-fuzzy quasi-coincident neighborhood spaces, which is called strong L-fuzzy quasi-coincident neighborhood space.     

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract

It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F _A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.     

Bredon homology and ramified covering G-maps

Let G be a finite group. The objective of this paper is twofold. First we prove that the cellular Bredon homology groups with coefficients in an arbitrary coefficient system M are isomorphic to the homotopy groups of certain topological abelian group. And second, we study ramified covering G-maps of simplicial sets and of simplicial complexes. As an application, we construct a transfer for them in Bredon homology, when M is a Mackey functor. We also show that the Bredon-Illman homology with coefficients in M satisfies the equivariant weak homotopy equivalence axiom in the category of G-spaces.     

Some of Melvin Henriksen?s contributions to spaces of ideals

The three of us have written this note to discuss Mel Henriksen?s joint paper with us, Joincompact spaces, continuous lattices andC^?-algebras. In this paper we learned that the space of closed primal ideals of a C^?-algebra is a continuous lattice, so it is joincompact when equipped with the lower and Scott topologies. In addition to its mathematics, we discuss how Mel brought us into this work, and his and its continuing influence on us.