Gabriele Castellini

Closure operators and connectedness

Abstract This paper introduces the notion of connectedness with respect to a closure operator on a construct X . Many classical results about topological connectednedness are extended to this setting. Herrlichs connectedness-disconnectedness Galois connection is shown to factor via the collection of all closure operators on X .

Closure Operators and Polaritiesa

ABSTRACT. Basic results are obtained concerning Galois connections between collections of closure operators (of various types) and collections consisting of subclasses of (pairs of) morphisms in 𝓂 for an (E, 𝓂)‐category 𝓂 In effect, the “lattice” of closure operators on 𝓂 is shown to be equivalent to the fixed‐point lattice of the polarity induced by the orthogonality relation between composable pairs of morphisms in 𝓂.

Connectedness with respect to a Closure Operator

A notion of connectedness with respect to a closure operator C and a class of monomorphisms N is introduced in an arbitrary category X. It is shown that under appropriate hypotheses, most classical results about topological connectedness can be generalized to this setting. Examples that illustrate this new concept are provided.

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.

CONNECTEDNESS, DISCONNECTEDNESS AND CLOSURE OPERATORS, A MORE GENERAL APPROACH

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 is introduced. This notion yields a Galois connection that can be seen as a generalization of the classical connectedness-disconnectedness correspondence (also called torsion-torsion free in algebraic contexts). It is shown that this Galois connection factors through the collection of all closure operators on X with respect to M).

A factorization of the Pumplün-Röhrl connection

Abstract The Galois connection given in 1985 by Pumplun and Rohrl between the classes of objects and the classes of morphisms in any category is shown (under ordinary circumstances) to have a “natural” factorization through the system of all idempotent closure operators over the category. Futhermore, each “component” of the factorization is a Galois connection in its own right. The first factor is obtained by using a generalization of the process, given by Salbany in 1975, that yields a closure operator for any class of topological spaces, while the second factor can be used to form the weakly hereditary core of an idempotent closure operator.

GLOBAL CLOSURE OPERATORS VS. SUBCATEGORIES: For the sixtieth birthday of Keith Hardie

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.

Interior operators and topological connectedness

Abstract A categorical notion of interior operator is used in topology to define connectedness and disconnectedness with respect to an interior operator. A commutative diagram of Galois connections is used to show a relationship between these notions and Arhangelskii and Wiegandts notions of connectedness and disconnectedness with respect to a subclass of topological spaces. Examples are included.

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.

Epimorphisms in categories of separated fuzzy topological spaces

Abstract The categorical theory of closure operators is used to characterize the epimorphisms in certain categories of separated fuzzy topological spaces (in the sense of Lowen). These include the 0 ∗ -T 0 -spaces of Wuyts and Lowen, the FT S -spaces of Ghanim, Kerre and Mashhour, and the α -T 2 -spaces of Rodabaugh.