When you try to understand the fuss behind the new good categories of spectra that arose on the 90's, you read things such as the following paragraph written by Peter May (from "The Hare and the Tortoise"):
All consumers are now in agreement: Mike's stable homotopy category is definitively the right one, up to equivalence. However, the really fanatical hare demands a good category even before passage to homotopy, with all of the modern bells and whistles. The ideal category of spectra should be a complete and cocomplete Quillen model category, tensored and cotensored over the category of based spaces (or simplicial sets), and closed symmetric monoidal under the smash product. Its homotopy category (obtained by inverting the weak equivalences) should be equivalent to Mike's original stable homotopy category.
Here "Mike" is Michael Boardman. Question number zero would be, could anybody share Boardman's "Stable homotopy theory" mimeographed notes where he introduces said category? Both that and Vogt's "Boardman's stable homotopy category" are hard to find, so it's hard for me to know what they are talking about.
But suppose I know what Boardman's stable homotopy category is. Why should I agree that this is "the" right stable homotopy category, up to equivalence?
I am guessing that the right way to formalize what I mean is: elabore a desiderata for a Stable Homotopy Category, prove that Boardman's satisfies them, and then prove that any two categories satisfying those axioms are equivalent.
Has that been done? If so, where?
I can try to answer my question. Margolis' book "Spectra and the Steenrod Algebra" from 1983 does have such a list of axioms, in section 1.2.
Is that an idiosyncratic list of axioms or is it really what homotopy theorists of the time would agree that it's exactly what they would have wanted?
I know this is perhaps argumentative. But the following is not. At the end of said chapter in Margolis' book, he conjectures that any two categories satisfying those axioms are equivalent. But at the time, it apparently wasn't established.
Has it been established since?
But maybe there is another characterization of "the" stable homotopy category, of which Boardman's (or Adams', or...) would be an example; I'd be interested by that, too.
I am mildly aware of the fact that there is a stable $\infty$-categorical universal property. That is certainly interesting, but I would be interested to see how it could be formulated in older language (model categories?) since such a formulation is, in the spirit of my question, anachronistic. (Not that I would find it uninteresting).