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.     

DISCONNECTEDNESS CLASSES

Abstract

Let X be an (E, M)-category for sinks. A notion of disconnectedness with respect to a closure operator C on X and to a class of X-monomorphisms N is introduced. This gives rise to the notion of N-disconnectedness class, a characterization of which is presented in a category with a terminal object. Some examples are provided.     

Factorizations,injectivity and compactness in categories of modules

A notion of closure operator for modules is used to characterize factorization structures in categories of modules. Moreover compactness, injectivity and absolute closedness are studied with respect to such closure operators. A criterion for compactness of modules is obtained in terms of injectivity or absolute closedness of the quotients extending recent results of Temple Fay.     

Categorical neighborhood operators

We introduce and study a concept of neighborhood operator on a category. Such an operator is obtained by assigning a suitably axiomatized stack of subobjects - the neighborhoods - to every subobject of each object in the category. We discuss closure and interior operators, convergence, separation and compactness with respect to a neighborhood operator, defined in a natural way.     

COMPLETIONS AND CATEGORICAL COMPACTNESS FOR NILPOTENT GROUPS

Abstract

We consider reflection functors in the category of nilpotent groups satisfying certain exactness properties for which the Mal'cev completion functor and the p-cotorsion completion functors are prototypical examples. Each of these functors defines a generalized torsion theory, which in turn defines a closure operator on subgroups. This gives rise to the notion of a categorically compact group with respect to the closure operator which we characterize. This approach provides a unified treatment for the categorically compact groups with respect to the Mal'cev completion and with respect to the p-cotorsion completion, the latter being new. We also consider the p-pro-finite completion, suitably restricted to obtain a reflection functor, and characterize the compact groups so arising.     

Closure operators and their middle-interchange law

The paper presents a new definition of closure operator which encompasses the standard Dikranjan-Giuli notion, as well as the Bourn-Gran notion of normal closure operator. As is well known, any two closure operators C, D in a category may be composed in, within order, two different ways. For a subobject M→X one may consider DX(CXM) or DCX(M)(M) as the value at M of a new closure operator D⋅C or D?C, respectively. The two binary operations are linked by a lax middle-interchange law. This paper explores situations in which the law holds strictly.     

<Emphasis Type="Italic">F</Emphasis>-Net Convergence with Respect to a Closure Operator

In a category with a closure operator, we introduce the notion of a neighborhood of a point. The neighborhoods are then discussed and used for defining a net convergence structure on the category with respect to the closure operator. The nets considered are obtained as a categorical generalization of the usual nets. We investigate basic properties of the defined convergence. In particular, we study convergence separation and convergence compactness and describe their relationships to the usual closure separation and closure compactness. Mathematics Subject Classifications (2000) 18D35, 54B30, 54A05, 54A20, 54D30.This research was supported by NATO-CNR Outreach Fellowship (No. 219.33) and by Grant Agency of the Czech Republic (No. 201/03/0933).     

Interior operators and topological separation

A notion of separation with respect to an interior operator in topology is introduced and some basic properties are presented. In particular, it is shown that this notion of separation with respect to an interior operator gives rise to a Galois connection between the collection of all subclasses of the class of topological spaces and the collection of all interior operators in topology. Characterizations of the fixed points of this Galois connection are given and examples are provided.     

An order-theoretic perspective on categorial closure operators

This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco’s allow one to see more easily the connections between standard conditions on general cco’s, and furthermore, we show that these connections for cco’s can be even deduced from the corresponding ones for bco’s, when considering cco’s relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco’s via bco’s is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco’s, independent results on cco’s are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco’s also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.     

Closure Operators with Respect to a Functor

A notion of closure operator with respect to a functor U is introduced. This allows us to describe a number of mathematical constructions that could not be described by means of the already existing notion of closure operator. Some basic results and examples are provided.     

Neighborhoods and convergence with respect to a closure operator

We study neighborhoods with respect to a categorical closure operator. In particular, we discuss separation and compactness obtained from neighborhoods in a natural way and compare them with the usual closure separation and closure compactness. We also introduce a concept of convergence based on using centered systems of subobjects, which naturally generalizes the classical filter convergence in topological spaces. We investigate behavior of the convergence introduced and show, among others, that it relates to the separation and compactness in natural ways.     

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.     

Compactness, Perfectness, Separation, Minimality and Closedness with Respect to Closure Operators

In this paper, the characterization of closed and strongly closed subobjects of an object in categories of various types of filter convergence spaces is given and it is shown that they induce a notion of closure. Furthermore, each of the notions of compactness, perfectness, separation, minimality and absolute closedness with respect to these two new closure operators are characterized in these categories and some known results are re-obtained.     

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.     

A categorical approach to absolute closure

Notions of strongly and absolutely closed objects with respect to a closure operator X on an arbitrary category X and with respect to a subcategory Y are introduced. This yields two Galois connections between closure operators on a given category X and subclasses of X, whose fixed points are studied. A relationship with some compactness notions is shown and examples are provided.     

The pullback closure operator and generalisations of perfectness

A categorical closure operator induced via pullback by a pointed endofunctor is introduced. Various notions of a perfect morphism relative to a pointed endofunctor and the induced closure are then considered. The main result explores how these notions are interrelated, linking also with earlier notions of perfectness.The author acknowledges financial support from the University of Cape Town, from the Foundation for Research Development through the Categorical Topology Research Group at the University of Cape Town, and from the University of L'Aquila.     

Protolocalisations of homological categories

A protolocalisation of a homological (resp. semi-abelian) category is a regular full reflective subcategory, whose reflection preserves short exact sequences. We study the closure operator and the torsion theory associated with such a situation. We pay special attention to the fibered, the regular epireflective and the monoreflective cases. We give examples in algebra, topos theory and functional analysis.     

CONNECTEDNESS AND DISCONNECTEDNESS: A DIFFERENT PERSPECTIVE

Abstract

A factorization of a Galois connection investigated earlier is used to give a definition of a connectedness-disconnectedness Galois connection that is free of the notion of constant morphism. A new notion of N-fixed morphism with respect to a class N of monomorphisms is presented. This is used to characterize the connectedness-disconnectedness Galois connection in the case that N is closed under the formation of pullbacks. Some closedness properties of these Galois connections are investigated.     

Neighborhoods with respect to a categorical closure operator

We introduce and study a concept of neighborhoods with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject lattices, naturally generalizes the classical neighborhoods in topological spaces and we show that it behaves accordingly. We investigate also separation and compactness defined in a natural way by the help of the neighboorhoods introduced.