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I am looking for a modern introduction to Spectra that improves on the treatment by Adams in his "Stable Homotopy and Generalized Homology" notes (by improves I mean taking into account what has been learned since the notes were written). In particular I'm interested in a source that covers some of the variations on Spectra (CW Spectra, Symmetric Spectra, Other types/categories of Spectra etc.) and Ring Spectra.

Question: What is good introduction to the modern point of view on Spectra?

I am particularly interested in the stable/unstable Adams' spectral sequence but the source need not take that as a goal.

As an aside I'll point out that notes from Hatcher in his unfinished book on spectral sequences has a short but nice, clear and concrete introduction to spectra. It does not go into the detail and depth I need.

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Obvious related question:mathoverflow.net/questions/70522/… –  B. Bischof Nov 23 '11 at 21:22
I've had a question like this in the back of my mind, but I've been too nervous to ask about it. If a reference like this doesn't exist, it sounds like there's a good amount of demand for someone to write one. –  Paul Siegel Nov 24 '11 at 14:08
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3 Answers

up vote 17 down vote accepted

[I'm a novice, and this got posted out of order: it answers Bak's question below.]

Sure, I can provide that. The cited reference was published in 1995, which was well before details of symmetric or orthogonal spectra were available, so it gives a fair amount of background but only refers to EKMM spectra for a modern category. There is a paper (Mandell, May, Schwede, Shipley) that compares all choices except EKMM, and there are various papers that compare those choices with EKMM, starting with a paper by Schwede. Those papers are maybe more technical than you want. A recent survey paper compares the various approaches philosophically: see Sections 11 and 12 of my paper

What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra? Geometry \& Topology Monographs 16(2009), 215--282.

That gives references and is fairly independent of Sections 1-10. It starts with a theorem (11.1) of Gaunce Lewis explaining that there is no ideal choice of category: if you assume your category has all the good properties you want, you reach a contradiction. The incompatibility comes when you ask for a homotopically meaningful symmetric monoidal structure on your category of spectra that also has a homotopically meaningful monoidal adjunction $(\Sigma^{\infty},\Omega^{\infty})$ relating spaces and spectra. I'm old-fashioned maybe, but I think spaces are still kind of important.

EKMM comes as close as possible to having such an adjunction, with the related advantage that all objects are fibrant and the related disadvantage that the sphere spectrum is not cofibrant. Symmetric and orthogonal spectra have the advantage that they are significantly easier to define and the sphere spectrum is cofibrant.
The simplicial version of symmetric spectra has the advantage that it is especially well-suited to adaptation to the motivic world. Orthogonal spectra have the advantage that they are much better suited for equivariant and parametrized generalizations than symmetric spectra. Common features are captured by the web of Quillen equivalences relating not just all known constructions but all possible ``good'' model categories of spectra: there is an axiomatization, due to Shipley; symmetric spectra play a privileged role in the proof.

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This is a great expansion on your previous comment. Exactly the kind of references I was looking for. Thanks! –  Anthony Bak Nov 29 '11 at 20:00
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There is probably no ideal source for this. The canonical choice for symmetric spectra is probably Stefan Schwede's book project http://www.math.uni-bonn.de/~schwede/SymSpec.pdf . There you will find a good treatment of symmetric spectra and especially ring spectra and comparision with other types of spectra. There is nothing in there about the Adams spectral sequence, but you probably know that the treatment of the Adams spectral sequence mainly relies on formal properties of spectra, which can be shown in any model and therefore Adams's treatment might still be one of the best.

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This is a great resource for Symmetric Spectra. Thanks. –  Anthony Bak Dec 1 '11 at 20:30
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Probably neither of these will be exactly what you're looking for, but here are two references that come to mind and might have some of what you want:

Algebraic Topology by Robert M. Switzer is a good classical source. It doesn't have the newer things you're looking for, but it's less hand-wavey than Adams tends to be.

On Thom Spectra, Orientability, and Cobordism by Yuli Rudyak. I don't remember exactly what's in there (probably not symmetric spectra), but I've found it to be a useful source in the past.

Certainly both of these handle ring spectra and module spectra.

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Greg, there are ring spectra and there are ring spectra: in the homotopy category and on the point-set level. The older sources like Switzer cannot possibly treat the modern world of point-set level ring and module spectra. In that old world, the cofiber of a map of module spectra is not a module spectrum: very unsatisfactory, and I'm sure not what Bak is looking for. One intentionally undetailed source is Modern foundations for stable homotopy theory'' (by EKMM authors) in I.M. James Handbook of algebraic topology''. –  Peter May Nov 24 '11 at 3:15
Peter, Can you elaborate? Are you recommending the "Modern Foundations for Stable Homotopy Theory" article (by you and others) because they are not detailed but give a good overview of the modern viewpoint? Since I'm not entirely sure what all the choices for definitions of spectra are (and what consequences such choices have) - is there something that lays out what the choices are and why you would chose one over another? –  Anthony Bak Nov 24 '11 at 5:29
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