What's the point of spectra?

I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and cohomology, but is this their sole purpose? I read in Switzer's Algebraic Topology book

"We have another aim in constructing the category of spectra. In homology theory the suspension homomorphism $\sigma: h_n(X) \rightarrow h_{n+1}( SX)$ is always an isomorphism. For various reasons this suggests trying to embed $\mathscr{PW}'$ in a larger category in which the suspension functor $S$ has an inverse $S^{-1}$."

Can anyone elaborate on this?

There are many different things going on here. One is that to invert the suspension functor is to stabilize. The Freudenthal suspension theorem tells you that the system of suspension maps $[X,Y] \to [SX,SY] \to [S^2X,S^2Y] \to \cdots$ eventually stabilizes (at least for finite CW-complexes), and so you can view this as a simplification of the usual category of spaces. Moreover, for formal reasons this implies that your category is enriched in abelian groups, which gives you more traction on things. (For instance, you can always add two maps, maps always have inverses, etc.)

More broadly, the real power of homotopy theory is that pretty much everything is representable in one way or another, which means that you can turn your tools on each other and get them to tell you new things about the tools themselves. From this point of view, it's entirely natural to define the category of spectra (since it is equivalent to the category of cohomology theories). But it sounds like you already knew about this piece of motivation.

• Hey Aaron! Be careful when saying "spectra are equivalent to cohomology theories," that's almost true but fails because of phantom maps. – Cary Dec 30 '13 at 3:41
• Thanks Cary! If you work with infinite CW complexes, though, I think this becomes alright again. A phantom map is a nontrivial map which can't be seen by any finite CW complexes. – Aaron Mazel-Gee Dec 30 '13 at 19:37
• @AaronMazel-Gee: One can also define morphisms of spectra that are phantom for all CW-complexes, not just finite, see for example mathoverflow.net/a/117768. – Dmitri Pavlov Mar 27 '15 at 18:30
• @DmitriPavlov Interesting, thanks for pointing that out! – Aaron Mazel-Gee Mar 27 '15 at 22:48

I am also learning all this, and I had the same crisis of asking what the hell are spectra for, so maybe I can help you with the idea that I have of spectra now. As Aaron said, there are many different things going on here, and his two reasons are already enough to motivate such a construction. Here are a few more (1reason and 2 good consequences).

I will assume that you believe that generalized cohomology theories are very useful in algebraic topology. Also, I am also learning this so take all I say with a grain a salt.

1) I think the main purpose and the motivation of why spectra are, is the following : there are some things that spaces hide to cohomology theories, and we would like to mod out by this "extra information" that we don't really need when we study spaces by means of cohomology theories. The information that spaces hide is the unstable phenomenon, in the following sense : if $X$ and $Y$ are stably equivalent, for example $\Sigma X \simeq \Sigma Y$, then $E_{\ast}(X) \cong E_{\ast + 1}(\Sigma X) \cong E_{\ast + 1}(\Sigma Y) \cong E_{\ast}(Y)$ for any generalized cohomology theory $E_{\ast}$. This says that there is no cohomology theory that is going to see a difference between $X$ and $Y$, so we might as well says that they are "the same". This is a reason why you would like to actually invert the suspension functor $\Sigma$, and the resulting (universal) category you obtain by doing so is the category of spectra. Another way to say this is that $Sp$ is the universal category with a functor $\Sigma^{\infty} \colon Top \to Sp$, such that any cohomology theory $Top \to GrAb$ factors trough it.

2) It turns out that this construction of spectra by inverting $\Sigma$ has many good properties, some of them are listed by Aaron. An important one (which is not true for spaces!) is that cofiber sequences are the same as fiber sequences. So we might as well try to work in this category $Sp$.

3) Another good consequence of spectra is Brown representability theorem. It says that any generalized cohomology theory on spaces is representable by a spectra. Spaces are not enough to get all the cohomology theories, you need a category a bit bigger than spaces. So now you can study cohomology theories with the tools you developed for studying spaces, nice. In particular, since a spectra = a cohomology theory, you can take the cohomology of a cohomology theory... Weird the first time you here about it !

• In #2, cofibrations and fibrations aren't quite "the same thing" -- this depends on taking a model for spectra (and in any model these are going to be inequivalent). Nor are pushouts and pullbacks the same thing. The important point here is that a pair of composable arrows is a fiber sequence iff it's a cofiber sequence. – Aaron Mazel-Gee May 4 '13 at 9:14
• @AaronMazel-Gee: thanks for the clarification. I edited the answer. – Bogdan May 11 '13 at 17:48
• When you say, this is a reason why you'd like to invert $\Sigma$ what exactly do you mean? Is it, if $\Sigma^{-1}$ exists then $\Sigma X \simeq \Sigma Y$ implies $X \simeq Y$? – user71210 May 16 '13 at 6:06
• @user71210: I mean something like this : since all (co)homology theories see the suspension functor $\Sigma$ as an isomorphism (since $E_{\ast}(X)$ and $E_{\ast}(\Sigma X)$ are the same, up to a shift by 1); it will make our life easier to declare that $\Sigma$ really is an isomorphism, because all (co)homology theories think it is. So let's formally invert$\Sigma$, similarly to how we form the fraction field of a domain by formally inverting all elements. This construction is called localization (of categories) and is usually not easy to construct. It turns out that Spectra is a model for ... – Bogdan May 18 '13 at 2:13
• ... this localized category. That is, the universal category with a functor $Top \to Spectra$ such that the suspension functor $\Sigma$ has now an inverse (the loop functor $\Omega$), in the same way that the fraction field (or a localization) is the universal field $R \to F$ such that everything is invertible. Furthermore, this category $Spectra$ is in fact a good model because as pointed out above, it is enriched in abelian groups, etc etc. Hope this makes more sense. – Bogdan May 18 '13 at 2:17

Aaron and redfiloux both have great answers here. I'd like to try to address the side of "these are basically cohomology theories" a little bit more carefully, hopefully without being too off-topic.

As an aside, a great reason to care about cohomology theories comes from Adams' two solutions to the Hopf invariant one problem. His first solution uses ordinary cohomology to do the work, but it is very long and makes extensive use of "higher cohomology operations." His second proof (with Atiyah) is beautiful and short, but only because he uses an extraordinary cohomology theory (complex K-theory) to do the job. (http://people.virginia.edu/~mah7cd/Foundations/Adams,%20Atiyah%20-%20K-theory%20and%20the%20Hopf%20Invariant.pdf)

So suppose we care about cohomology theories. These are fairly simple on their own, they just turn each space into a sequence of abelian groups. Certainly we can get pretty far by just calculating these groups and using them to prove theorems.

Only for a given theorem, we may need to construct a new cohomology theory out of older ones. For instance, we may need to take a pushout or direct limit of cohomology theories. This is actually quite difficult if you think of cohomology as a functor from spaces to groups. You can make a category of such functors, but this category does not contain all colimits, so you are left without a means of "gluing together" cohomology theories to make new ones.

You may be familiar with a similar problem at the space level. One can construct the "homotopy category of spaces" by taking the objects to be CW complexes and the morphisms to be homotopy classes of continuous maps. This is a category, all right, but it does not contain pushouts or sequential colimits. So you can work with spaces and maps-up-to-homotopy if you like, but you won't be able to do much. It's much better to work with spaces and maps on-the-nose, and to make constructions like the double mapping cylinder and the mapping telescope when you want to form pushouts or sequential colimits. A good mantra is, pass to homotopy as late as possible.

The same reasoning applies to spectra vs. cohomology theories. The stable homotopy category of spectra (as described by Adams and Switzer) is almost equivalent to the category of cohomology theories (but not quite: https://mathoverflow.net/questions/117684/are-spectra-really-the-same-as-cohomology-theories). We can take double mapping cylinders (and homotopy cofibers, etc.) of spectra, and this gives us a meaningful way of "gluing together cohomology theories." The extra topology involved (a sequence of spaces instead of a sequence of groups) gives us more control and a greater ability to make constructions like this.

Hopefully that provides some motivation to study spectra!