Talk:Spectrum (topology)

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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credits Boardman with "[T]he first satisfactory construction of the stable homotopy category" in his (I believe) still unpublished paper of 1964 (page 48).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)

What is $Z$?

What is $Z$? —The preceding unsigned comment was added by 217.189.225.6 (talk • contribs).

Z_(disambiguation)#Mathematics - all integers. Not sure if this is obscure enough to clarify in the article. (John User:Jwy talk) 18:46, 17 June 2007 (UTC)

Rewrite...

Hi,

I'm trying to rewrite this somewhat sprawling text into something more cohesive. Also, the plan is to talk about coordinate-free genuine spectra as being different from (pre)spectra and emphasise the difference. There's a draft at User:Rswarbrick/Scratchpad, which I intend to work on over the next few days before merging. If anyone thinks this is a terrible idea, now would be a good time to tell me! Of course, if anyone wants to jump in, feel free to edit the draft: I'd love help! Rswarbrick (talk) 15:05, 17 November 2010 (UTC)

Expert attention needed

Sorry, but some fairly recent and well-intentioned editing has introduced real problems. The page may appear more readable, but the errors involved are substantial enough to make it misleading. One or two of them involve convention as well as correctness, so it's important to have an expert writer better than me. For details, please see history of this talk page.

Ambrose H. Field (talk) 21:34, 23 October 2011 (UTC)

Suspension shift and spectra indexed by natural numbers

There is a flaw in the article: a spectrum $X$ is indexed by natural numbers, so is has spaces $X_{0}$, $X_{1}$, etc. The suspension of $X$ is now the spectrum $\Sigma X$ with 0-th space $X_{1}$, and $X_{0}$ is simply thrown away. This will not yield an invertible functor.

One way out is to allow spectra to be indexed over the integers, but bounded below: there exists some natural number $N$ such that $X_{n}=\emptyset$ for all $n\leq -N$.

This problem should be at least pointed out in the article. Konrad (talk) 14:42, 14 May 2012 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Spectrum (topology)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

* This article is a self-assessed stub, for which I can give the following reasons: it doesn't have any indication of the development of the subject, in particular the modern development (which it asserts is significant); there are only two examples, together with a cop-out "example" pointing to the list of cohomology theories (rather than actually giving more example of spectra per se), and although the examples illustrate the subject they don't really give a window into its nature; no references are given (but there are several discussed on the talk page that could be added); the only real attempt to address the question "what are spectra" is a definition, which is not enlightening. Someone should try incoprorating the ideas thrown about (nearly two years ago!) on the talk page. Ryan Reich 01:46, 6 July 2007 (UTC)

Substituted at 21:58, 26 June 2016 (UTC)

"Smash products of spectra" section

The section Smash products of spectra does not bother to explain what a "smash product of spectra" means.

This is exactly what an encyclopedia article should not be. If the article is going to mention a new concept, it should at minimum say what that concept means.

Otherwise, readers are left with the name of a new concept, but no concept for it. If you want to write for Wikipedia, please take the trouble to say what you are talking about. Otherwise you do more harm than good.50.205.142.50 (talk) 13:26, 12 May 2020 (UTC)

I agree this sections needs work (but I disagree, that harm is done to a reader by this. Nothing is factually wrong there, but it needs context and elaboration). Since you seem to care about the topic, please be bold and contribute to this article! Jakob.scholbach (talk) 15:00, 12 May 2020 (UTC)

Killing and truncation

There should be a section discussing killing elements in spectra and truncating the ring spectrum. In particular, the examples on page 78 of http://www.math.uni-bonn.de/~schwede/SymSpec.pdf should be included and be used as guiding light. Converting the relevant material into a referenced article would be useful as well.