13 - Large worlds of mathematical structures

Summary

We explore other large categories based on the category of sets and functions. We begin with monoids, and think of them as sets with extra structure. We introduce the concept of a structure-preserving function, which in this case gives us the definition of a monoid morphism. We check that we can assemble monoids and their morphisms into a category. We do the analogous construction for groups and group homomorphisms. We consider partially ordered sets and show how the structure-preserving functions in this case are order-preserving functions, and we assemble those into a category. We consider some cases of functions that are not order-preserving, illuminating some antagonism that can arise over the theory of privilege. We present the category of topological spaces and continuous maps. We introduce the idea of assembling categories themselves into a category, beginning to think about the idea of structure-preserving maps for categories. These are called functors, and are covered in full in a later chapter. Finally, we touch on matrices, introducing a category where the objects are natural numbers and a morphism from a to b is an a × b matrix. This concludes the interlude of the book.