FACTORIZATIONS OF CONES ALONG A FUNCTOR

Abstract

First a general Galois correspondence is established, which generalizes at the same time the correspondence between classes of monomorphisms and injective objects and the correspondence between classes of epimorphisms and monomorphisms in a category. This correspondence arises naturally if one tries to generalize some concepts of “topological” or also of “algebraic” functors. Both kinds of functors admit certain factorizations of cones, and just this fact implies some of their common nice properties: lifting limits, continuity and faithfulness, for instance. These properties can be shown without having a left adjoint. Therefore the theory yields also applications to functors which are neither “topological” nor “algebraic”.     

ON CERTAIN FACTORIZATIONS OF FUNCTORS INTO THE CATEGORY OF QUASI-UNIFORM SPACES

This paper is motivated by the search for natural extensions of classical uniform space results to quasi-uniform spaces. As instances of such extensions we restate some theorems of P. Fletcher and W.F. Lindgren [Pacific J. Math. 43 (1971), 619–6311 on transitive quasi-uniformities and of S. Salbany [Thesis, Univ. Cape Town, 1971] on compactification and completion. The theorems as restated describe properties of certain right inverses of the functor which forgets the quasi-uniform structure and retains one induced topology (for Fletcher and Lindgren's work), respectively retains both induced topologies (for Salbany's work). Accordingly we investigate systematically the process by which the right inverses of the forgetful functors can be extended from the classical setting to one of these settings, and from one of these to the other.     

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.     

On categorical notions of compact objects

Due to the nature of compactness, there are several interesting ways of defining compact objects in a category. In this paper we introduce and study an internal notion of compact objects relative to a closure operator (following the Borel-Lebesgue definition of compact spaces) and a notion of compact objects with respect to a class of morphisms (following Áhn and Wiegandt [2]). Although these concepts seem very different in essence, we show that, in convenient settings, compactness with respect to a class of morphisms can be viewed as Borel-Lebesgue compactness for a suitable closure operator. Finally, we use the results obtained to study compact objects relative to a class of morphisms in some special settings.Partial financial assistance by Centro de Matemática da Universidade de Coimbra and by a NATO Collaborative Grant (CRG 940847) is gratefully acknowledged.     

Strong inclusion orders between L-subsets and its applications in L-convex spaces

Abstract

In this paper, the concept of strong inclusion orders between L-subsets is introduced. As a tool, it is applied to the following aspects. Firstly, the notion of algebraic L-closure operators is proposed and the resulting category is shown to be isomorphic to the category of L-convex spaces (also called algebraic L-closure spaces). Secondly, restricted L-hull operators, as generalizations of restricted hull operators, are introduced and the resulting category is also proved to be isomorphic to the category of L-convex spaces. Finally, by using the properties of strong inclusion orders, it is shown that the category of convex spaces can be embedded in the category of stratified L-convex spaces as a reflective subcategory and the concrete form of the coreflective functor from the category of L-convex spaces to the category of stratified L-convex spaces is presented.     

ALGEBRAIC HULLS OF ADJOINT FUNCTORS AS INJECTIVE HULLS

Abstract

For each adjoint functor U: A → X where X is an (?, M)-category having enough ?-projectives, we construct an (?, M)-algebraic hull E: (A, U) → (Â, Û), i.e., (Â, Û) is (epsiv; M)-algebraic and E has a certain denseness property. We show that there is a conglomerate of functors over X with respect to which the (? M)-algebraic categories are exactly the injective objects and characterize (? M)-algebraic hulls as injective hulls.     

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 inherits from a category the property that all descent morphisms are effective if either is regular and is a full coreflective, closed under pullbacks of certain epimorphisms, subcategory of or is regular, has coequalizers and there exists a topological functor . 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.     

Zariski closure, completeness and compactness

The categorical theory of closure operators is used to introduce and study separated, complete and compact objects with respect to the Zariski closure operator naturally defined in any category X(A,Ω) obtained by a given complete category X (endowed with a proper factorization structure for morphisms) and by a given X-algebra (A,Ω) by forming the affine X-objects modelled by (A,Ω). Several basic examples are provided.     

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.     

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.     

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.     

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.     

SOME MODULES OVER A B*-ALGEBRA

Abstract

Some modules, ?_p (B) (1 < p < ∞), over a commutative B*-alge= bra with identity are discussed and these are seen to have properties similar to the Banach spaces ?_p (1 < p < ∞).     

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].