CATEGORIES OF TOPOLOGICAL GROUPS

Abstract

This paper is a survey of recent (and some not so recent, results concerning categorical constructions on topological groups, with particular emphasis on free topological groups and coproducts (free products) of topological groups. An extensive bibliography is included.     

ON EPIDENSE SUBCATEGORIES OF TOPOLOGICAL CATEGORIES

Dense subcategories were introduced by S. Marde?i? for an inverse system approach to (categorical) shape theory.

In this paper some internal characterizations of (epi,bi)dense subcategories of a topological category are given. We also show that if K ? A is a bidense subcategory then the “best approximation” of an A-object X by a K-inverse system is obtained by “modifications” of the structure of X.     

T0-SEPARATION IN TOPOLOGICAL CATEGORIES

R.-E. Hoffmann [5,6] has introduced the notion of an (E,M)-universally topological functor, which provides a categorical characterization of the T₀-separation axiom of general topology. In this paper, we characterise these functors in terms of the unique extension of structure functors defined on the subcategory of “separated” objects (of the domain category). This, in turn, leads to a solution of some problems due to G.C.L. Brümmer [1,2]. Other results include a generalization of L. Skula's characterization of the bireflective subcategories of Top [10].     

TWO FAMILIES OF CARTESIAN CLOSED TOPOLOGICAL CATEGORIES

Abstract

In this paper two ordered families of topological categories are studied. The first family includes the category of all abstract simplicial complexes and the subcategories of all abstract simplicial complexes of dimension less than or equal to n. The categories of the second family are bireflective subcategories of the category of all bornological spaces. All these categories are cartesian closed and have other nice properties.     

CONNECTEDNESS,DISCONNECTEDNESS AND CLOSURE OPERATORS: FURTHER RESULTS

Abstract

Let X be an arbitrary category with an (E, M)-factorization structure for sinks. A notion of constant morphism that depends on a chosen class of monomorphisms was previously used to provide a generalization of the connectedness-disconnectedness Galois connection (also called torsion-torsion free in algebraic contexts). This Galois connection was shown to factor through the class of all closure operators on X with respect to M. Here, properties and implications of this factorization are investigated. In particular, it is shown that this factorization can be further factored. Examples are provided.     

EPIREFLECTIVE SUBCATEGORIES AND SEMICLOSURE OPERATORS

ABSTRACT

Let C be a category of topological spaces and continuous functions which is full, hereditary and closed under homeomorphisms and products. If A is a subclass of C, let E(A) be the full subcategory of C whose objects are the subspaces in A. In this paper we characterize the epireflective subcategories of C containing A and contained in E(A) by introducing a “semiclosure” operator which is a generalization for the “idempotent semi-limit” operator introduced by S.S. Hong (see [5]) with respect to Top _o. In case A is extensive in C, so that E(A) = C, all the extensive subcategories of C containing A are thus characterized.     

SOME RECENT RESULTS ON CATEGORIES OF R-SPACES AND PROBABILISTIC SPACES AND ON CATEGORICAL SPECTRA

The purpose of this paper is twofold: first, to present some recent results obtained by Romanian mathematicians in the field of general and categorical topology; second, to present some current research results obtained by the author in what may be called the topological study of a category. Accordingly, the paper is divided into two parts.

The author wishes to express his gratitude to the organizing committee of this Symposium for the kind invitation to present' this paper.     

AN AXIOMATIC APPROACH TO CATEGORIES OF TOPOLOGICAL ALGEBRAS

Abstract

Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:

1. if h: UA → UB and g: VA → VB are morphisms with Gh = Tg, there exists a morphism f: A → B such that Uf = h and Vf = g;

2. V carries U-initial monosources into T-initial mono-sources.

The functor V has many nice properties which shed light on the blending of the “topology” and “algebra”; e.g., V is a topologically algebraic functor in the sense of Y.H. Hong. An ([Etilde],[Mtilde]) version of O. Wyler's “Taut Lift Theorem” is used to show that the existence of a left adjoint to V is related to Condition (ii). It is also shown that certain topological algebraic reflections arise as Topological Algebraic Situations from algebraic and topological surjective reflections.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.     

GLOBAL CLOSURE OPERATORS VS. SUBCATEGORIES

Given a concrete category (A,U) over a category X, we establish a Galois correspondence between subcategories of A and global closure operators over A.     

EQUI-MORPHIC FAMILIES IN CATEGORIES

Motivated by his previous work on proximally fine and on equi-p-fine uniform spaces, the author extends some results about equi-uniformly continuous families of functions to the general setting of equi-morphic families in a category.     

Quotient-reflective and bireflective subcategories of the category of preordered sets

In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T₀, T₁, and T₂ preordered sets and show that each of the full subcategories of each of T₀, T₁, T₂ preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord.     

Hausdorff separation in categories

Considering subobjects, points and a closure operator in an abstract category, we introduce a generalization of the Hausdorff separation axiom for topological spaces: the notion ofT ₂-object. We discuss the properties ofT ₂-objects, which depend essentially on the behaviour of points, and finally we relate them to the well-known separated objects.The results of this paper are essentially taken from the author's Ph. D. Thesis written under the supervision of Professors M. Sobral and W. Tholen and partially supported by a scholarship of I.N.I.C.-Instituto Nacional de Investigação Científica.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?_j|dπ^j is defined, where {?_j:j = 1,…, n} denotes an invariant basis of the planes of the distribution D_n. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution D_n is said to be pseudo-Lagrangian if dπ^j(?_j,V_h) = 0; if D_n, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π² ? ? πⁿ. If π is closed, the distribution D_N is integrable, and the exterior system {π^j} admits the representation ψ^j = dS^j in terms of a set of 0-forms S^j on M. If, in addition, the distribution D_N is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution D_n, be both canonical and integrable. The last of these requirements implies the existence of N functions ψ^A that depend on x^h and N parameters w^B, whose derivatives are given by ?ψ^A (x^h, w^B)/?x^j = B^A _j (x^h, ψ^B (x^h,w^B)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψ^A and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function.     

Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions

In non-symmetric Convenient Topology the notion of pre-Cauchy filter is introduced and the construction of a precompletion of a preuniform convergence space is given from which Wyler's completion of a separated uniform limit space [O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970) 1-12] as well as Weil's Hausdorff completion of a separated uniform space [A. Weil, Sur les Espaces à Structures Uniformes et sur la Topologie Générale, Hermann, Paris, 1937] can be derived (up to isomorphism). By the way, the construct PFil of prefilter spaces, i.e. of those preuniform convergence space which are ‘generated’ by their pre-Cauchy filters, is a strong topological universe filling in a gap in the theory of preuniform convergence spaces.     

A NOTION OF COMPLETENESS OF TOPOLOGICAL STRUCTURES

A connector U on a space S is a function from S to the power set of S such that each x in s belongs to its image. The image of x is denoted by xU. In other words, the relation {(x,y): y ? xU, x ? S) is a reflexive binary relation. A space with a certain set of connectors is a generalization of topological spaces as well as uniform spaces. In this paper, a notion of completeness of such a space is introduced. This completeness corresponds to completeness of uniform spaces if a set of cannectors meets the conditions of uniformity. Compactness of topological Spaces is a special case of the completeness.     

A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π:X×Y→Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X×Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article.Categorical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.     

BIFRAMES AND BISPACES

Abstract

The concept of a biframe is introduced. Then the known dual adjunction between topological spaces and frames (i.e. local lattices) is extended to one between bispaces (i.e. bitopological spaces) and biframes. The largest duality contained in this dual adjunction defines the sober bispaces, which are also characterized in terms of the sober spaces. The topological and the frame-theoretic concepts of regularity, complete regularity and compactness are extended to bispaces and biframes respectively. For the bispaces these concepts are found to coincide with those introduced earlier by J.C. Kelly, E.P. Lane, S. Salbany and others. The Stone-?ech compactification (compact regular coreflection) of a biframe is constructed without the Axiom of Choice.