A graduate-level textbook that presents basic topology from the perspective of category theory.
This graduate-level textbook on topology takes a unique approach: it reintroduces basic, point-set topology from a more modern, categorical perspective. Many graduate students are familiar with the ideas of point-set topology and they are ready to learn something new about them. Teaching the subject using category theory—a contemporary branch of mathematics that provides a way to represent abstract concepts—both deepens students' understanding of elementary topology and lays a solid foundation for future work in advanced topics.
After presenting the basics of both category theory and topology, the book covers the universal properties of familiar constructions and three main topological properties—connectedness, Hausdorff, and compactness. It presents a fine-grained approach to convergence of sequences and filters; explores categorical limits and colimits, with examples; looks in detail at adjunctions in topology, particularly in mapping spaces; and examines additional adjunctions, presenting ideas from homotopy theory, the fundamental groupoid, and the Seifert van Kampen theorem. End-of-chapter exercises allow students to apply what they have learned. The book expertly guides students of topology through the important transition from undergraduate student with a solid background in analysis or point-set topology to graduate student preparing to work on contemporary problems in mathematics.
This book is at the leading edge of what will likely become a major pedagogical trend in mathematics: teaching the fundamentals from a categorical perspective.
David Spivak, Research Scientist at MIT, author of Category Theory for the Sciences
As an algebraic topologist who has taught point-set topology from an implicitly category-theoretic viewpoint for many years, I was delighted to discover this beautifully written textbook.
Kathryn Hess, Professor, EPFL
Bradley, Bryson, and Terilla make a compelling case for approaching category theory through point-set topology, imparting a lovely point of view that enlivens both subjects.
Emily Riehl, Associate Professor, Johns Hopkins University, author of Categorical Homotopy Theory and Category Theory in Context
The categorical approach used is not only well motivated, but presented in a style that is very user-friendly.
Jim Stasheff, Professor Emeritus UNC-CH; Visiting Researcher at the University of Pennsylvania; coauthor of Characteristic Classes