# Different point-set level definitions of spectra

I've been trying to understand the Adams spectral sequence and one of the more accessible sources is the (unfinished) book on spectral sequences by Hatcher. The usual (only?) construction of the spectral sequence uses spectra and so in the book you can find an extremely brief introduction to the theory.

Hatcher defines a spectrum to be a sequence of basepointed spaces $E_{n}$ together with connecting maps $\sigma _{n}: \Sigma E_{n} \rightarrow E_{n+1}$ and it seems that in fact this is the most common definition. As I understand, this leads to the right notion of objects but the right notion of morphisms between spectra is more subtle. To work around this, one introduces the more strict CW-spectra which are sequences of CW-complexes with connecting maps inclusions of subcomplexes. As Frank Adams shows in his classic work "Stable homotopy and generalised homology', this approach - together with slightly modified notion of morphism and homotopy of morphisms - leads to well-behaved stable homotopy category.

However, for example in May's "A Concise Course in Algebraic Topology", the above notion of spectrum is downgraded to a prespectrum and one might expect that, again, spectra are somewhat more special. This can be seen in the paper "Modern Foundations for Stable Homotopy Theory" (by Elmendorf, Kriz, Mandell, May), where - although in a different guise - the authors define a prespectrum $E_{n}$ to be a spectrum if the adjoint maps $E_{n} \rightarrow \Omega E_{n+1}$ are homemorphisms. Of course, there is much more going on in the paper, as they are trying to obtain a category of spectra with reasonable smash product, but already at the beginning this homeomorphism condition puzzles me, because it seems extremely strong. I also don't know what is the correct notion of a "map" in this setting since the paper becomes rather demanding very quickly, but one may expect some model structure to play a role?

Which of these approaches to a point-set definition of spectra are the most well-established? What are their advantages and disadvantages?

(Right now I'm mostly concerned with the "additive" properties of spectra, ie. I already know there are even more approaches when one tries to construct a category of spectra with a smash product and this is not really the main point of my question.)

• If you don't have the right morphisms, you don't have the right objects. – Qiaochu Yuan Jun 10 '13 at 19:47
• You are, of course, right, since a category is as good as is morphisms. I was just making a little nod to how one might pass to a category with the "right morphisms" (without changing the objects) by - let's say - some process of localization. – Piotr Pstrągowski Jun 10 '13 at 23:01
• If you're interested in the Adams spectral sequence then I guess you only really care about the homotopy category, rather than the point set level constructions. You can pick any model for spectra (S-modules, orthogonal/symmetric spectra) and when you pass to the associated homotopy category, they all give the same thing. If you really do want a point-set construction, perhaps you should look at Schwede's book on symmetric spectra: math.uni-bonn.de/~schwede/SymSpec.pdf – Drew Jun 10 '13 at 23:09

## 1 Answer

A lot of exciting things have been done with the foundations of spectra in the last 20 years, but it's not so clear where to start when you're learning the subject.

To fix notation, let's call the sequence of spaces and structure maps $\Sigma E_n \rightarrow E_{n+1}$ a prespectrum. A map of prespectra $E \rightarrow F$ is just a sequence of maps $E_n \rightarrow F_n$ which commute with the structure maps. Nothing fancier than that. This defines the category of prespectra. A map of prespectra is a $\pi_*$-isomorphism if it induces isomorphisms on all the stable homotopy groups.

Finally, since the category described by Adams has different objects and different morphisms than the one I described above, we'll give it a different name: the stable homotopy category.

As you suggest in your comment above, if you take the category of prespectra and localize at the class of $\pi_*$-isomorphisms, you get (up to equivalence) the stable homotopy category. Even better, you can put a model structure on the category of prespectra whose weak equivalences are the $\pi_*$-isomorphisms, and the associated homotopy category is also equivalent to the stable homotopy category. I believe this was first worked out in a paper of Bousfield and Friedlander, but you can find a nice modern treatment in the paper Model Categories of Diagram Spectra (Mandell, May, Schwede, Shipley), which also extends to symmetric/orthogonal spectra. If I remember correctly, when you work with strict $\Omega$-spectra you instead restrict to a class of "CW spectra" and then take honest homotopy classes of maps between them to recover the stable homotopy category. See Equivariant Stable Homotopy Theory (Lewis, May, Steinberger) for more details. (It's also a good prequel to EKMM.)

As for your main question, there are many well-established models for spectra: prespectra, coordinate-free $\Omega$-spectra, symmetric spectra, orthogonal spectra, $\Gamma$-spaces... I have found the category of prespectra with the Bousfield-Friedlander model structure to be an excellent starting point. Here are three reasons to focus on that category:

• The definitions are simple so it's easy to get your hands dirty.
• You can do just about anything that doesn't require smashing spectra together or forming mapping spectra.
• Most other models have an underlying prespectrum by neglect of structure, so when you move on to a different model you can build on what you already know.

Try learning the definitions, and then proving that homotopy cofiber sequences and homtopy fiber sequences coincide. (Hint: start by proving that cofiber and fiber sequences both yield long exact sequences of stable homotopy groups.) Once you've built some intuition there, you may be drawn toward symmetric/orthogonal spectra, since their definitions are pretty simple. Or you may be drawn toward $\Omega$-spectra, which have good formal properties and which leverage classical work on the geometry of infinite loop spaces. At any rate, I think you'll find that time spent understanding prespectra is not wasted when you move on to more sophisticated models.