Introducing Category Theory

NB: A revised version of this second edition is forthcoming in early 2026.

Modern pure mathematics explores abstract structures. Such structures cluster in interrelated families which form structures of structures. And these higher level structures are in turn interconnected in intricate ways. How can we explore such layers of increasing abstraction without getting lost? Category theory provides a basic tool-kit, and it throws very revealing light on ideas which recur across mathematics.

This book is a based on a much-downloaded set of notes, and aims to give an introduction to some core categorial concepts. Part I looks inside categories, giving general treatments of familiar constructions such as forming products, quotients, exponentials, and more. Part II explores the functors that can map a construction in one category to the same sort of construction in another category, and we eventually encounter some distinctive novelties of category theory, such as the Yoneda Lemma and the concept of adjunctions. Part III looks more briefly at one kind of category, the elementary toposes in which we can reconstruct much mathematics.

The pace is gentle, with many theorems set as ‘challenges’ which are then given worked out proofs. So the book will provide very accessible preliminary or parallel reading for those starting a course on category theory. It will also be of interest to anyone who wants to get some sense of what the categorial fuss is about, as the book presupposes relatively little mathematical background.

Before he retired from the University of Cambridge, Peter Smith taught logic for more years than he cares to remember. His books include An Introduction to Formal Logic (2003, 2020), An Introduction to Gödel’s Theorems (2007, 2013), Gödel Without (Too Many) Tears (2020, 2022) and Beginning Mathematical Logic: A Study Guide (2022). He was also editor of Analysis for a dozen years.