Topology Book Launch
This is the official launch week of our new book, Topology: A Categorical Approach, which is now available for purchase! We're also happy to offer a free open access version through MIT Press at* topology.mitpress.mit.edu.
Inside, you'll find a presentation of basic, point-set topology from the perspective of category theory, targeted at graduate students in a first-semester course on topology. The idea is that most of these students are already somewhat familiar with the point-set ideas through a course on analysis or undergraduate topology. For this reason, many graduate-level instructors are tempted to rush through point-set topology or to skip it altogether to reach algebraic topology, which can be more fun to learn and to teach.
Our book presents an alternative to this approach. Rather than skipping over the basic ideas, we view this as an excellent opportunity to introduce students to the modern viewpoint of category theory. As we note in the preface,
We think this alternative is better for several reasons. Since many students are familiar with point-set ideas already, they are in a good position to learn something new about these ideas, like the universal properties characterizing them.
Plus, using categorical methods to handle point-set topology, whose name even suggests an old fashioned way of thinking of spaces, demonstrates the power and versatility of the methods. The category of topological spaces is poorly behaved in some respects, but this provides opportunities to draw meaningful contrasts between topology and other subjects and to give good reasons why some kinds of spaces (like compactly generated weakly Hausdorff spaces) enjoy particular prevalence.
Finally, there is the practicality that point-set topology is on the syllabus for our first-year topology courses and PhD exams. Teaching the material in a way that both deepens understanding and prepares a solid foundation for future work in modern mathematics is an excellent alternative.
To accomplish this, you'll find a number of categorical gems inside the text:
- a presentation of subspaces, quotient spaces, products, and coproducts in terms of universal properties (chapter 1)
- a functorial proof of Brouwer's fixed point theorem (chapter 2)
- an introduction of filters in lieu of sequences in the discussion on convergence (chapter 3)
- a proof of Tychonoff's theorem using ultrafilters (chapter 3)
- a thorough discussion of limits and colimits (chapter 4)
- a categorical presentation of the compact-open topology, with historical comments (chapter 5)
- discussions on homotopy, the fundamental groupoid, adjunctions, and fibrations, all with an eye towards homotopy theory (chapter 6)
- and more!
Not covered are covering spaces, homology, and cohomology, but readers will be well-equipped for these topics after reading through our text. In particular, the preface lists some excellent resources on where to pick up on algebraic topology afterwards.
Thanks for your support!
A number of you have already purchased a copy of the book, and the shipments are starting to roll in. Many thanks for sharing your photos and support! We sincerely appreciate all the positive feedback you've given.
And by the way, you might notice a difference in writing style between what's on Math3ma and what's in the book. Indeed, the audiences for these two mediums are quite different. Even so, if you have questions while embarking on the text, don't forget to check out the topology and category theory libraries here on the blog! I wrote most of those articles while the book was in preparation. In fact, you'll find a few of my favorite "golden nuggets" on Math3ma hidden throughout.
As a final thought, our book will be used as the main text for this semester's topology course at the CUNY Graduate Center. As with most universities, classes will be held online this semester, and so a collection of videos may be created to accompany our book. Stay tuned for this!
In the mean time, enjoy!