An anonymous source told me this question is here. Dylan gave the quick answer and Tyler referred to it.

I'll use the question as an excuse to give a pontificating longer answer. When I first planned on writing that, maybe 45 or 50 years ago, I had not yet been converted to model category theory, let alone anything more modern, and what I had in mind would have been very plodding. I've thought hard about the pedagogy, perhaps not to good effect, and Concise and More (or less) Concise give what I came up with. The latter is not as concise as I would like in large part because so much relevant detail seemed missing from both the algebraic and topological foundations of localization and completion, especially in the proper generality of nilpotent rather than simple spaces. Both are available on my web page.

http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

http://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf

As an historical note, in retrospect I came to the conclusion that the standard foundations for classical algebraic topology, as I understood it a half century ago, were given by the mixed model structure on spaces discovered by Mike Cole. The weak equivalences are the same as in the usual Quillen model category, but the fibrations are the Hurewicz fibrations rather than the Serre fibrations. Then the cofibrant spaces are the spaces of the homotopy types of CW complexes, which for sure was everybody's favorite category of spaces in which to work back then.