Notes on pre-frölicher Spaces

Abstract

In this work we introduce a class of Sikorski differential spaces (M;D) called pre-Frolicher spaces, on which the process of yielding a Frolicher structure on the underlying set M is D preserving, their category we denote by preFrl. We investigate some algebraic properties on these spaces whose subsequent geometric properties are mostly similar to those of smooth manifolds, except for the invariance of dimension, and also that preFrl naturally induces a Cartesian closed subcategory of the category Frl in which there is no discrete object. Using this Cartesian property, it is shown that the Gelfand representation is a smooth map, that the tangent as well as cotangent bundles are made smooth spaces in an unusual but more natural way via smooth curves.

Mathematics Subject Classication (2010): 20G05, 46M15, 53B40, 54C40, 57R22.

Key words: Differential space, Frolicher space, pre-Frolicher space, cotangent bundle,
functor, Gelfand representation.