On the multitude of monoidal closed structures on UAP

Abstract

In this note, we prove that all compact Hausdorff topological spaces are exponential objects in the category UAP of uniform approach spaces and contractions as introduced in R. Lowen, Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad, Oxford University Press, 1997. As a consequence, we show that UAP admits at least as many monoidal closed structures as there are infinite cardinals. We also prove that under the assumption that no measurable cardinals exist, there exists a proper conglomerate of these monoidal closed structures on UAP.

On some special classes of continuous maps

“…Exponentiable objects in approach theory are studied in [21,14], and the following sufficient condition is obtained.…”

Exponentiable approach spaces

“…This fact triggers the question of finding sufficient and necessary conditions for an approach space X to be exponentiable, that is, for the cartesian product functor (−) × X : App → App to have a right adjoint. A first important result in this direction was obtained in [Lowen and Sioen, 2004] where it is shown that every compact Hausdorff spaces is exponentiable in the category of uniform approach spaces and contractions. Two years later, [Hofmann, 2006] presents a sufficient condition motivated by the characterisation of exponentiable generalised metric spaces obtained in Clementino and Hofmann [2006] (see Theorem 4.4).…”