One Smooth Category to Rule Them All

In the land of differentiability, smooth manifolds have yielded a rich and successful theory, spawning fields of mathematics and moving into other sciences as well. However, as a category, smooth manifolds are quite limited. For example, singular varieties and orbit spaces are often not manifolds at all. In this talk, we will examine different generalisations of "smoothness", building an increasingly complex Venn diagram of "categories of smooth objects" as we greedily search for one category that encompasses all abstract structures we could possibly want. The talk is meant to be light, with motivating examples to illustrate the benefits of these generalisations.