George E. Strecker

A Primer on Galois Connections

ABSTRACT. The rudiments of the theory of Galois connections (or residuation theory, as it is sometimes called) are provided, together with many examples and applications. Galois connections occur in profusion and are well known to most mathematicians who deal with order theory; they seem to be less known to topologists. However, because of their ubiquity and simplicity, they (like equivalence relations) can be used as an effective research tool throughout mathematics and related areas. If one recognizes that a Galois connection is involved in a phenomenon that may be relatively complex, then many aspects of that phenomenon immediately become clear, and thus, the whole situation typically becomes much easier to understand.

Galois connections and computer science applications

We have presented an existence theorem and some important properties of Galois connections. We have also shown how data structures problems can be simplified and better understood when Galois insertions are used. In particular, the proof of correctness of an implementation follows simply from the construction of a Galois insertion. We plan further applications of Galois connections theory to computing-related problems.

Factorizations, denseness, separation, and relatively compact objects

Abstract Some of the relationships among the topological notions: ‘Hausdorff’, ‘compact’, ‘perfect’, and ‘closed’ are abstracted to a more general categorical setting, where they are shown to remain intact. An investigation is made of factorization structures (especially for single morphisms) and their relationships to strong limit operators and to the Pumplu¨n-Ro¨hrl Galois correspondence between classes of objects and classes of morphisms in any category. Many examples as well as internal characterizations of Galois-closed classes are provided.

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.

A Category of Galois Connections

We study Galois connections by examining the properties of three categories. The objects in each category are Galois connections. The categories differ in their hom-sets; in the most general category the morphisms are pairs of functions which commute with the maps of the domain and codomain Galois connections. One of our main results is that one of the categories—the one which is the most closely related to the closed and open elements of the Galois connections—is Cartesian-closed.

The compactness operator in general topology

This chapter discusses the compactness operator in general topology. The role of (bi)compactness has increased tremendously during the past half century. The chapter focuses on the strengthening of this notion at the expense of the Hausdorff property. It discusses a few special cases of importance such as: (1) ϱ = ɛ holds exactly for those topological spaces in which the compact sets coincide with the closed sets. (2) ϱ 2 = ɛ. In this case ϱ and ɛ form a group of order 2 with ɛ as the identity. Spaces supplied with such a minus topology are called antispaces. These are exactly those spaces in which the square-compact subsets coincide with the closed subsets. The locally compact Hausdorif spaces and the metrizable spaces are for example antispaces.

Lagois connections: a counterpart to Galois connections

Abstract In this paper we define a Lagois connection, which is a generalization of a special type of Galois connection. We begin by introducing two examples of Lagois connections. We then recall the definition of Galois connection and some of its properties; next we define Lagois connection, establish some of its properties, and compare these with properties of Galois connections; and then we (further) develop examples of Lagois connections. Via these examples it is shown that, as is the case of Galois connections, there is a plethora of Lagois connections. Also it is shown that several fundamental situations in computer science and mathematics that cannot be interpreted in terms of Galois connections naturally fit into the theory of Lagois connections.

Categorical Topology — Its Origins, as Exemplified by the Unfolding of the Theory of Topological Reflections and Coreflections before 1971

“Man is a being, intelligent and gifted with the faculty of comprehending the abstract. Thanks to this faculty, man has conceived the ideal, and realized poesy; he has conceived the infinite, and created mathematics. Such is the immense distinction which separates the human race so widely from the animals, which makes him a being apart and absolutely new upon the globe. Comprehending the ideal and the infinite, creating poetry and algebra, such is man! To find and understand this formula

Injectivity of topological categories

{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}