# Modern source for spectra (including ring spectra)

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 a 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.

• Obvious related question:mathoverflow.net/questions/70522/… Nov 23, 2011 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. Nov 24, 2011 at 14:08

## 5 Answers

[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.

• This is a great expansion on your previous comment. Exactly the kind of references I was looking for. Thanks! Nov 29, 2011 at 20:00

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.

• This is a great resource for Symmetric Spectra. Thanks. Dec 1, 2011 at 20:30

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.

• 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''. Nov 24, 2011 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? Nov 24, 2011 at 5:29

Urs Schreiber has written beautiful notes on Stable Homotopy Theory. These can be found here . He starts from very basic level, and goes all the way upto Adams spectral sequences. I have studied the 'Prelude' and 'Part 1', and found the notes to be very clearly written. (In fact I wish all Mathematics books/notes were written as clearly as these.) Hope you find these helpful.

I have found Barnes and Roitzheim's book Foundations of Stable Homotopy Theory to be a good introduction. I'll quote my review from when it first came out.

FOUNDATIONS fills in an important missing piece in the homotopy theory literature: an introduction to stable homotopy in modern language which doesn't assume any previous knowledge of the subject. Barnes and Roitzheim approach stable homotopy theory from the point of view of model categories, which allows them to discuss stable phenomena in an appropriately broad context without overwhelming the unfamiliar student. This also allows for a clearer exposition of the theory that is particular to topological spectra, which is more easily understood and separated from the general theory with said general theory already taken care of. The logical structure of the text is generally superb, with well-established motivation and a clear "story" to follow through each chapter and through the text as a whole (notwithstanding a few references to future chapters which can usually be taken on faith). The necessary model categorical background is contained in the appendix, but it is quite bare-bones and often skimps on proof; I would recommend learning the basics from Dwyer and Spalinski's classic paper and referring to the appendix and its references when necessary.

There are only two issues I have with the text. The first issue is the errors. FOUNDATIONS is a massive text currently in its first edition, so it is natural that it has some errors in it, but they are frequent enough as to be occasionally confusing. Moreover, while most of the errors will be noted and easily fixed by the attentive reader, there are two or three proofs that need more serious revision. This will no doubt become less of a problem as time goes on, with errata being written and future editions being published, but it is something that the prospective learner should be aware of. The second issue, which is more philosophical, is that the text is light on computation and examples. Apart from a section on the Steenrod algebra and another giving a brief exposition of the Adams spectral sequence, FOUNDATIONS spends almost all of its time on developing the theory, with very few explicit examples. Computational stable homotopy theory is a subject in of itself, and fitting it into this book would require entire new chapters, but some demonstrations of the constructions and techniques the text develops would go a long way in aiding understanding.

Overall, I would highly recommend this book for a beginner in stable homotopy theory. Just make sure to read it with a mentor or advisor who can help you understand the more difficult sections, compensate for the relative dearth of examples, and spot errors.