# An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. right from the start.

I'm comfortable with general topology and category theory, but I haven't had much exposure to algebraic topology beyond the basics of the fundamental group(oid) and de Rham cohomology. In particular, I'd like to learn about the various homology and cohomology theories.

• I highly recommend Tammo Tom Dieck's book, which takes a categorical approach even if it doesn't always explicitly use categorical language.
– user314
Commented May 25, 2015 at 13:45
• I like Tom Dieck's book as well,but I think it's too sophisticated for any but the best students. To effectively read it,one needs to be VERY comfortable with basic algebra( up to and including R-modules and basic category theory) in addition to some background in point set topology, the fundamental group and homotopy theory. I think this is a tall order for most beginning graduate students in the U.S.I also think it's too dry and abstract,completely disconnecting the material from it's geometric sources.In a sense,it's the "anti-Hatcher".May has the same problem,only not as bad. Commented Sep 13, 2016 at 5:52

Rotman's An Introduction To Algebraic Topology is a great book that treats the subject from a categorical point of view. Even just browsing the table of contents makes this clear: Chapter 0 begins with a brief review of categories and functors. Natural transformations appear in Chapter 9, followed by group and cogroup objects in Chapter 11.

The aspect I like most about this book is that Rotman makes a clear distinction between results that are algebraic and topological. E.g., he proves several statements about group actions before then applying them to the particular topological setting of covering spaces and the action of the fundamental group on a fiber.

• +1 for what I think is the best "abstract" text on algebraic topology for beginners. Students that master Rotman will be well-equipped to move on to more sophisticated monographs and research papers. Rotman is one of our very best textbook authors. Commented Sep 13, 2016 at 5:42

Spanier's book is relatively old (so I know it does not quite answer your question), but excellent. It uses category theory from the get-go. Riehl's "Categorical homotopy theory" is very well-written, though it may be a bit too advanced if you hadn't seen a bit of algebraic topology already. Riehl's book is focused on the categorical aspect via Quillen model structures. A major and important area of algebraic topology. The categorical requirements for appreciating the Quillen machinery are not modest, and the book does a great job presenting all that is needed.

Having said all that, if you find a text that you like but which avoids the category theoretic language, then you should be able to quite easily fill-in the gaps on your own. Simply consult online sources (e.g., the nLab) to get the categorical pictures (and then some) of whatever concept you are learning. In an introductory text you will probably cover the fundamental group(oid), Van Kampen's Theorem, some higher homotopy groups, and some homology. It should be easy to find the categorical/functorial description of these concepts no matter how the book presents it.

• Filling in "on your own" isn't always so easy; language dictates how people think, and it isn't all that uncommon to see a text have tell you things about products, but remain silent about the corresponding statement for equalizers, so filling in the gap requires you to actually develop that part of the theory yourself (or discover relevant counterexamples on your own).
– user14972
Commented May 25, 2015 at 6:28

I suggest Peter May's Concise course on algebraic topology. You will find e.g. categorical formulations (and proofs) of the van Kampen theorem and the classification of covering spaces.

• Thanks! I think this is just what I was looking for.
– ಠ_ಠ
Commented May 26, 2015 at 23:05

The closest thing I've found is Strom's Modern Classical Homotopy Theory, although I haven't read much of it. Chapter 1 is called Categories and Functors, so that's a good start. This is the only introductory algebraic topology textbook I know of that explicitly uses the language of homotopy limits and colimits.

• I have to mention that the notion of the fundamental groupoid on a set of base points has been well supported by A. Grothendieck in his 1984 "Esquisse d'un Programme" but will be found in texts in English on topology only in the book "Topology and Groupoids" -do a web search (though that book contains no homology). . Commented Mar 6, 2018 at 15:35