Historical transition from classical homotopy to modern homotopy theory

Question

I plan to begin seriously studying model categories and their 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 he 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?

Answer 1

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. André–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 Strøm'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.

Answer 2

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 question, I find "nice expensive edition" an oxymoron. One does not want yet another "prohibitively expensive" source. It would be a service to get Whitehead's classical book scanned and made available on line. (See, May is not quite that old-fashioned.) Sean, thanks for the plug, but [90-92] on my web site are all stable. Lennart, thanks too for your plug "\ul{here}", which is to [110] on my web site. But that is hard. Thanks more for your advertisement of Cole's mixed model structures, which we agree is where classical homotopy theory actually lives. [116] on my website (with Ponto) is an advanced but hopefully readable "modern" textbook source that advertises that point of view in its treatment of model categories.

Answer 3

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 Mahowald. 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.

The nice thing about this is that many of the players are still alive and active.

EDIT: I had forgotten about the following fun resources: I.M. James was the editor of the Handbook of Algebraic Topology and History of Topology. The first has an article on each "new" area, the second goes through... the history. The first is prohibitively expensive, but most of the articles are available on the websites of the authors.