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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.
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.
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.
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.
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.
