TITLE: An introduction to algebraic topology from the categorical point of view
QUESTION [13 upvotes]: 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.

REPLY [5 votes]: 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.