Talk:Spectrum (homotopy theory)

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With regard to the introduction of spectra and Boardman: according to J. P. May, Stable Algebraic Topology, 1945-1966 (available here in a variety of formats), spectra were first introduced in 1959 by E. I. Lima, a student of G. W. Whitehead (page 20). May credits Boardman with "[T]he first satisfactory construction of the stable homotopy category" in his (I believe) still unpublished paper of 1964 (page 48). Reference: Lima, The Spanier-Whitehead duality in new homotopy categories. Summa Brasil Math. 4 (1959), 91-148.

Alodyne 00:10, 6 Dec 2004 (UTC)

It would certainly be a help to have a page on S-duality. I tend to know these things from the Frank Adams perspective.

Charles Matthews 05:58, 6 Dec 2004 (UTC)

I am about to indulge in a bit of philosophical rambling. Some of this will eventually make it onto the article page, when I get it coherent in my mind. Please comment as you like.

"Making the suspension invertible" is indeed a very common motivation in texts and courses for the introduction of the objects we call spectra. But to me, this is less convincing than representability of all cohomology functors (see Brown representability theorem). The entire topic is difficult to treat. As I imply in my earlier comment, there are many different constructions of various categories, all of which purport to be "the category of spectra" or "the stable homotopy category." Adams' construction is described in the article, and I think that this is still the model that most homotopy theorists have in mind. But Boardman's model has better formal properties. The really cutting edge categories are better still (there are two that I'm aware of: E-infinity ring spectra and S-modules) because they are model categories with good properties before passage to homotopy, most notably a smash product that commutes on the nose. But do we really want to talk about any of this in the article? My worry is really that it is misleading to define a spectrum as is done in the article, from the modern viewpoint; but I can't see that any alternative is really any better.

On another note, I think the comparison to derived categories might be delivered with less apology. There is quite a bit of current research happening along those lines. For example, there are various sets of axioms one might use to define a stable homotopy category, e.g., triangulated, closed symmetric monoidal, a set of categorically small generators, etc. With suitable choices the derived category satisfies these. See the monograph of Hovey, Palmieri, and Strickland, Axiomatic Stable Homotopy Theory. Moreover the model categories approach applies here as well. Consider the category of bounded chain complexes of modules. The cofibrant objects in the usual model structure are the perfect complexes, the ones with each term projective (I think?). So projective resolution is cofibrant replacement, and this is actually well-defined up to unique isomorphism (I think?) in the derived category.

Any thoughts?

Dave Rosoff 04:10, July 23, 2005 (UTC)

What is clear to me that is there is not a single category of spectra, but any reasonable category of spectra should at least yield the same stable homotopy category. I recommend covering not only the classical definition of spectrum here, but also subsequent definitions. These could be compared with each other and to the original definition, making for a useful and educational article. Spectra nowadays don't even have to be topological in nature: spectra of simplicial sets have been defined, and there is an extensive theory that also gives a model for stable homotopy theory. On the other hand, more advanced constructions such as symmetric spectra would probably require their own article for an adequate explanation (by saying this I am not volunteering to write one ;-)). - Gauge 05:42, 16 December 2005 (UTC)

Mostly that I put this up from the Switzer book on algebraic topology; and I'm aware that it lacks technical detail. Some more of that would be very good. On the general aspect: what WP likes is accurate history (which I think in this case is what Boardman did?), plus reliable current references, plus definitions at the point that we can have them. What does that exclude, then? Well, we are not really supposed to go for the optimal, ideal treatment where that exceeds the textbooks. Charles Matthews 16:27, 27 July 2005 (UTC)