Categorical Perspectives

Frontmatter

Categories: A Free Tour

Abstract

Category theory plays an important role as a unifying agent in a rapidly expanding universe of mathematics. In this paper, an introduction is given to the basic definitions of category theory, as well as to more advanced concepts such as adjointness, factorization systems and cartesian closedness.

Lutz Schröder

The Functor that Wouldn’t be

A Contribution to the Theory of Things that Fail to Exist

Y. T. Rhineghost, H. Herrlich

The Emergence of Functors

A Continuation of “The Functor that Wouldn’t be”

Y. T. Rhineghost, H. Herrlich

Too Many Functors

A Continuation of “The Emergence of Functors”

Alois Zmrzlina

Contributions and Importance of Professor George E. Strecker’s Research

Abstract

We give an overview of the long and distinguished career of Professor George E. Strecker in the fields of topology and, in particular, categorical topology.

Jürgen Koslowski

10 Rules for Surviving as a Mathematician and Teacher

Abstract

I am indeed honored that a gathering was organized to celebrate my sixtieth birthday. When I turned forty, Horst Herrlich told me that at that point I was entitled to wax philosophical¹. So now I feel that I am more than entitled — I’m actually obliged to do so. So here goes…

George E. Strecker

Connections and Polarities

Abstract

Galois connections - also called residuated/residual maps - have been studied and used extensively in both mathematics and computer science. In the 1980s, Galois connections were generalized to connections, and in the 1990s, a counterpart to Galois connections, called Lagois connections, were discovered. Lagois connections, as the name suggests, are similar to Galois connections; however, the “movement” from arbitrary points to image points - which is done via the composite maps of the connections - is in the same “direction” with respect to the order relations in both partially ordered sets. This similarly directed movement seems to be characteristic of (many) computer science applications. In this chapter we present these concepts in their “discovery” order; we also give properties and examples. We begin with a pre-Galois connection concept, called a polarity.

Austin Melton

Categorical Closure Operators

Abstract

A brief survey of the development of the theory of closure operators is presented. Results concerning the applications of the theory to epimorphisms, separation, compactness and connectedness are also included together with a number of supporting examples.

G. Castellini

Extensions of Maps from Dense Subspaces

Abstract

In 1971, Douglas Harris introduced a concept that he called a WO-map in order to find a subcategory of the category of T ₁ topological spaces for which the Wallman compactification becomes functorial. Later, Bentley and Naimpally generalized Harris’ result to the setting of topological spaces endowed with the structure of a separating base in the sense of Steiner. In the present paper, the generalization is carried one step further to the setting of nearness spaces in the sense of Herrlich. Thus, it is shown that by restricting the class of maps, but not the class of spaces, the (strict) completion of a nearness space becomes functorial.

H. L. Bentley

Characterizations of Subspaces of Important Types of Convergence Spaces in the Realm of Convenient Topology

Abstract

Since in Convenient Topology we are mainly concerned with semiuniform convergence spaces, the question arises how the subspaces of important types of convergence spaces such as topological spaces, pretopological spaces (=closure spaces in the sense of Čech [5]), limit spaces (in the sense of Kowalsky [10] and Fischer [6]) or Kent convergence spaces can be characterized when they are considered as semiuniform convergence spaces (provided all convergence spaces fulfill a certain symmetry condition). This paper presents the solution. Furthermore, the relationships to other important subconstructs of the construct SUConv of semiuniform convergence spaces are investigated.

Gerhard Preuß

The Naturals are Lindelöf iff Ascoli Holds

Abstract

It is shown that in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the classical Ascoli Theorem holds iff ℕ is a Lindelöf space.

Y. T. Rhineghost, H. Herrlich

Revisiting the Celebrated Thesis of J. de Groot: “Everything is Linear.”

Abstract

By a system (X, T) we mean a continuous selfmap T: X → X of a separable metrizable space X and by its linearization a topological embedding i: X → E into a linear system (E, L) consisting of a Hilbert or Euclidean space E and a continuous linear operator L: E → E and satisfying the equivariancy condition L ○ i = i ○ T. Our main results concern linearization by systems (E, L) in which the norm of L is < 1. By a weakening of the equivariancy condition L ○ i = i ○ T, we show that a system which does not admit a finite dimensional linearization may still be linearized in this modified sense in a finite dimensional space.

Ludvik Janos

Finite Ultrametric Spaces and Computer Science

Abstract

The purpose of the paper is to describe a few properties of ultrametric spaces (in particular, of finite ones) and to demonstrate some applications of these properties to computer science.

A metric space (X, d) is called ultrametric [6] (or non-Archimedean [4], or isosceles [9]) if its metric satisfies the strong triangle axiom:

$$ d\left( {x,z} \right) \leqslant \max \left[ {d\left( {x,y} \right),d\left( {y,z} \right)} \right]. $$

(Δ)

This is usually called the Ultrametric Axiom. Ultrametric spaces were described up to homeomorphism in [3, 21], up to uniform equivalence in [10], and up to isometry in [9, 20]. A survey of their metric [9, 20], geometric [14, 20], uniform [10], and categorical [11–17] properties can be found in the literature. The theory of ultrametric spaces is closely connected with various branches of mathematics. These are number theory (rings Z _p and fields Q _p of p-adic numbers), algebra (non-Archimedean normed fields), real analysis (the Baire space \( {B_{{\aleph _o}}} \)), general topology (generalized Baire spaces B _τ ), p-adic analysis (field Ω), p-adic functional analysis (algebras of Ω-valued functions), lattice theory [17], Lebesgue measure theory [18], Euclidean geometry [14], category theory and topoi [13, 15, 16], and so on. These relations deal with infinite ultrametric spaces (mainly separable). For applications in computer science, finite spaces are of interest as well.

Vladimir A. Lemin

The Copnumber of a Graph is Bounded by [3/2 genus (G)] + 3

Abstract

We prove that the copnumber of a finite connected graph of genus g is bounded by [3/2g]+3. In particular this means that the copnumber of a toroidal graph is bounded by 4. We also sketch a proof that the copnumber of a graph of genus 2 is bounded by 5.

Bernd S. W. Schröder

Abelian Groups: Simultaneously Reflective and Coreflective Subcategories versus Modules

Abstract

We investigate full subcategories of the category Ab of Abelian groups that are simultaneously reflective and coreflective in Ab. Such subcategories are exactly those isomorphic to categories of modules that are fully embedded into Ab. Rings giving rise to such modules are completely described. One of the curious special cases is provided by the full subcategory of Ab consisting of all torsion-free, divisible Abelian groups, which can be characterized, alternatively as the reflective hull of ℚ in Ab, or as the coreflective hull of ℚ in Ab, or as the intersection of the epireflective hull of ℚ in Ab with the monocoreflective hull of ℚ in Ab.

Robert El Bashir, Horst Herrlich, Miroslav Hušek