How Many Closed Structures does the Construct PRAP Admit?

Abstract

We will prove that the topological construct PRAP, introduced by E. and R. Lowen in [9] as a numerification supercategory of the construct PRTOP of convergence spaces and continuous maps, admits a proper class of monoidal closed structures. We will even show that under the assumption that there does not exist a proper class of measurable cardinals, it admits a proper conglomerate (i.e. one which is not codable by a class) of mutually non-isomorphic monoidal closed structures. This severely contrasts with the situation concerning symmetric monoidal closed structures, because it is shown in [13] that PRAP only admits one symmetric tensorproduct, up to natural isomorphism.