# Historical transition from classical homotopy to modern homotopy theory…

I plan to begin seriously studying model categories and thier applications to homotopy theory this summer. But I was hoping the topologists and historians in here could help me with something related:I was hoping to begin a push to get George Whitehead's ELEMENTS OF HOMOTOPY THEORY republished in a nice expensive edition for students. But looking at Whitehead's opus, it's amazing how alien most of it looks compared to the model categoric approach used today-long calculations with complexes and spectral sequences-and many problems were simply too difficult to attack directly.

My problem is if this work was republished, there really should be some historical context attached to it so students could transition from it to the more abstract methods today. (Sadly, Whitehead himself was planning a second volume detailing the model categorical approach-which was just beginning to become widely used in research at that point-and apparently he gave up attempting to compose it before be passed away.) Does anyone know a good historical account of the transitional works between classical homotopy theory and the modern approach? I was hoping Whitehead's own "50 Years Of Homotopy Theory" would do the job and it would be perfect to bookend with the treatise,but it's not really about that. None of the review articles on model categories-like Dwyer,et.al.-really do this either.

Can anyone outline historically this development for me?

-
Algebraic topology is too big to have a coherent history. The bomb dropped in 1945 when Eilenberg and MacLane wrote the "book" and we have been reacting ever since. –  Charlie Frohman May 2 '10 at 22:48
One point to make is that Whitehead's book wasn't necessarily the standard approach to teaching homotopy theory when it was published, either, particularly the amount of time devoted to spectral sequences. (See Adams' review at projecteuclid.org/euclid.bams/1183545223). It is also not obsolete. It simply has a different emphasis (e.g. many modern texts don't devote much time the classical Lie groups). –  Tyler Lawson May 3 '10 at 1:52
@KConrad, those were AndrewL's words, not Yemon's. –  S. Carnahan May 3 '10 at 5:29
"Algebraic topology is too big to have a coherent history." What? Entire civilizations have "coherent histories" that can be communicated in meaningful and useful ways. Surely algebraic topology, and even all of mathematics, being infinitely less complex, is amenable to similar treatment? –  Pietro KC May 3 '10 at 8:42
I'd just like to point out that model categories don't really offer an 'alternative approach' to homotopy theory. They are an axiomatization of the techniques that had already been developed for working with spaces or simplicial sets, and the idea was to use them as a guide for applying the same yoga in other settings. –  Jeffrey Giansiracusa May 3 '10 at 8:47

I am surely not a historian of topology, but I might try a few words.

That the usual literature concerning model categories is quite far away from traditional homotopy as presented in Whitehead's classic, is no wonder. Indeed, model categories are abstracted from homotopy theory, but not really that of the classical flavour. Quillen's lecture notes are not without reason entitled 'Homotopical Algebra'. As discussed in its introduction, its main object is to present an abstract framework where one can consider simplicial objects in categories of relevance for algebra. This leads to a theory of "non-additive derived functors", e.g. Andre-Quillen homology.

In particular, the example of the model structure on topological spaces inducing the classical homotopy category is not presented in Quillen's book - only the one using Serre fibrations, generalized CW-complexes and weak homotopy equivalences. The model structure with Hurewicz fibrations/cofibrations and homotopy equivalences had to wait until Strom's The homotopy category is a homotopy category . As a consequence, the first absorbers of the theory of model categories were more simplicial minded guys. See for example Bousfield and Kan's Homotopy limits, completions and localizations.

One reason, why the notion of a model category is today so omnipresent in algebraic topology is that they provided a very good framework to discuss the homotopy theory of spectra and it was important to work both simplicially and in topological spaces. But the model structure on topological spaces used here was again the Quillen model structure.

I think, it is only in the last years that topologists are caring more again to reunion classical homotopy theory and model categories. One important work for this is Cole's Mixing model structures . Here, a model structure on topological spaces is constructed, where the weak equivalences are again the weak homotopy equivalences, but fibrations are now the Hurewicz fibrations. This leads to a theory, where the cofibrant objects are all spaces homotopy equivalent to a CW-complex. This model structure interacts rather well with more classical homotopy theory (using Hurewicz cofibrations and so on) as is seen e.g. here or in (section 8 of) this, which is also used in the five-author paper Units of ring spectra and Thom spectra. The reason, why the latter needs the connection to more classical homotopy theory is that the theory of $E_\infty$-spaces stems from classical homotopy theory and is simultaneously deeply linked to modern stable homotopy theory.

-
Thank you very much!Your response was very enlightening and when I can get out of this hole I've dug for myself before final exams,I'll be sure to check out those cited works. –  The Mathemagician May 3 '10 at 16:09

I was re-reading sections of Whitehead's book the other day, and I found it very helpful to think in the way he was writing. For a historical perspective, I would ask Clarence Wilkerson, Peter May, Bill Dwyer, Stewart Priddy, Dan Kan, and Mark Mahalwold. Among those, I expect that Peter May is closest to the transitional point of view. I think of Mike Hopkins as on the modern side of the fence.