Comparative Smootheology

TL;DR

This paper compares various definitions of the category of smooth objects by constructing functors between them, revealing their relationships within a unified framework involving test object categories.

Contribution

It introduces a general method to compare different smooth categories by embedding them into a broader context with test objects, unifying their relationships.

Findings

Categories of smooth objects are related through constructed functors.

The framework applies beyond smooth spaces to other categories involving test objects.

Fr"olicher spaces serve as a central reference point in the comparison.

Abstract

We compare various different definitions of "the category of smooth objects". The definitions compared are due to Chen, Fr\"olicher, Sikorski, Smith, and Souriau. The method of comparison is to construct functors between the categories that enable us to see how the categories relate to each other. This produces a diagram of categories with the category of Fr\"olicher spaces sitting at its centre. Our method of study involves finding a general context into which these categories can be placed. This involves considering categories wherein objects are considered in relation to a certain collection of standard test objects. This therefore applies beyond the question of categories of smooth spaces.