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Akhil, as one of those who struggled, I'd like to point out that nobody in a million years would have first come up with such an esoteric construction (if it is one) of the smash product of spectra, a …

Pages 75-81 of Appendix A of On the theory and applications of differential torsion products, Memoirs AMS 142 (1974), by V.K.A.M. Gugenheim and myself, gives a detailed treatment of the $W$-constructi …

This is not an answer, but a little too long for a comment. Scott, what a list: 3 dead, 2 retired, and me; also I think Mike and I are always on the same side of the fence. As to the original questi …

A 2020 paper Operads, monoids, monads, and bar constructions by Ruozi Zhang, Foling Zou, and J.P. May revisits the foundations of operad theory and discusses several possible choices for what $\mathca …

I'm not quite certain what Peter May had in mind 40 years ago, but probably he had in mind the fact that pushouts are a lot better behaved in CGWH than in CGH. Specifically, CGWH is closed under push …

I see no real point in considering $E_{\infty}$ spaces with $0$ except in the context of the multiplicative structure of an $E_{\infty}$ ring space or, essentially equivalently, the multiplicative str …

A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space …

I'd like to expand on Dustin's point. There is simply no way to think sensibly about equivariant topology, whether algebraic or geometric, without taking account of multiple basepoints. Even taking …

There are two old papers that address this topic in some detail: R. W. Thomason. Uniqueness of delooping machines. \url{https://projecteuclid.org/euclid.dmj/1077313403} Z. Fiedorowicz. Classifying …

There is so much much more. For one historical starting point among many, you see that many spaces of interest are infinite loop spaces and that tells you how to calculate things about them. For jus …

Notice the switch in grading from homology to cohomology on May's page 182: $P^s(x) = P_{-s}(x)$. Operations that raise degree when graded homologically lower degree when graded cohomologically. A la …

I'll take your question as license to advertise a relatively recent paper in a slightly more specialized but concretely calculational direction: http://nyjm.albany.edu/j/2014/20-53p.pdf. Its title …

I'm feeling mischievous. The history of the Kervaire invariant problem is strewn with false proofs, Jim Milgram's is only one of many. A student of mine (who I will leave nameless) had a preprint (a …

@FKranhold You mean I got it right? You had me fooled. I should apologize for leaving that detail to the reader, but let me give two excuses. First, one does not actually need that detail to prove L …

I have no idea what source you are quoting, but you are quite right that it is wrong, unless we are both screwing up. One builds $\phi_n$ rigorously by inducting on the stages of the construction of …