The idea of focusing on the relationships between mathematical objects, rather than on their internals, is fundamental to modern mathematics, and category theory is the framework for working from this perspective.
**Introduction**. Traditionally, the first chapter of a textbook on mathematics begins by recalling basic notions from set theory. This chapter begins by introducing basic notions from category theory, the shift being from the internal anatomy of sets to their relationships with other sets. The idea of focusing on the relationships between mathematical objects, rather than on their internals, is fundamental to modern mathematics, and category theory is the framework for working from this perspective. Our goal for chapter 0 is to present what is perhaps familiar to you—functions, sets, topological spaces—from the contemporary perspective of category theory. Notably, category theory originated in topology in the 1940s with work of Samuel Eilenberg and Saunders MacLane (Eilenberg and MacLane, 1945).
This chapter’s material is organized into three sections. Section 0.1 begins with a quick review of topological spaces, bases, and continuous functions. Motivated by a few key features of topological spaces and continuous functions, we’ll proceed to section 0.2 and introduce three basic concepts of category theory: categories, functors, and natural transformations. The same section highlights one of the main philosophies of category theory, namely that studying a mathematical object is akin to studying its relationships to other objects. This golden thread starts in section 0.2 and weaves its way through the remaining pages of the book—we encourage you to keep an eye out for the occasional glimmer. Finally, equipped with the categorical mindset, we’ll revisit some familiar ideas from basic set theory in section 0.3.