Topology: A Categorical Approach
I've been collaborating on an exciting project for quite some time now, and today I'm happy to share it with you. There is a new topology book on the market! Topology: A Categorical Approach is a graduate-level textbook that presents basic topology from the modern perspective of category theory. Coauthored with Tyler Bryson and John Terilla, Topology is published through MIT Press and will be released on August 18, 2020. But you can pre-order on Amazon now!
Here is the book's official description:
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.
In addition to the hard copy, MIT Press will also have a free online HTML version! I'll share more details about this and the contents of the book as we get closer to the release date. In the mean time, I hope you'll spread the word! I'm certainly excited to share this with you. [Update, Aug 18, 2020: The online book is now available here.]
It's hard to keep a good secret!
By the way, the cover art is called "Hopf Fibration" and is the work of mathematical artist London Tsai. I first heard of Tsai's extraordinary work on Eric Weinstein's podcast The Portal, and instantly became a fan. Many thanks to London!