Comparative Smootheology

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.