Visualising Spectra?

Question

I have recently started to learn about Spectra. To state the definition (that I have learned), a spectrum $X = \{X_{n} \}_{n \geq0}$ is a sequence of based spaces $X_{n}$, with basepoint preserving maps $\Sigma X_{n} \rightarrow X_{n+1}$ for all $n \geq 0$.

Maps between CW spectra may then be defined in terms of cofinal subspectra (c.f. Hatcher's Chapter 5 SSAT); homotopies between maps of CW spectra can be defined in terms of a map from the cylinder object of the domain to the range (c.f. Adam's book); you can go on and define homotopy and (co)homology etc.

My question is whether if there are any 'enlightening' pictures I can have in my mind when thinking about the above?

For example, when thinking about usual spaces, I think of the fundamental group in terms of equivalence classes of loops (and higher homotopy groups as a 'generalisation' of this); I think of (integral) homology as counting the number of 'holes' in a space and homotopy equivalent spaces as spaces that can be 'collapsed' from one to another (to mention a few examples).

I understand this is a bit of a soft question, but I suppose I am looking for some pictures that can deepen my understanding of spectra.

Any thought on this would be much appreciated!

Answer 1

As a philosophy, you can think of stable phenomena as what happens if you suspend everything enough times. That makes it hard (for me at least) to have any geometric intuition — what does $S^{35}$ look like? — but it also means that you should perhaps start with suspension spectra: those with $X_n = \Sigma^n X$ for a fixed space $X$. Then you can think of the suspension spectrum of $X$ just as the space $X$, or maybe $X$ suspended a few times.

A separate philosophy is that the homotopy category of spectra behaves in some ways like vector spaces: there is a smash product (= tensor product), there is a duality functor which is well-behaved on finite spectra, and more generally the set $F(X,Y)$ of homotopy classes of maps from $X$ to $Y$ has the structure of a spectrum and interacts well with the tensor/smash product.

Answer 2

This is a couple years late, but I figure it can't hurt to provide a different perspective for other readers. One way of visualizing spectra is to think in terms of cell diagrams, which were probably used in some form or another by many homotopy theorists going back a few decades. However, the source I typically recommend to people if they really want to understand computations with spectra is the Kervaire invariant paper by Barratt, Jones, and Mahowald. There is a section explaining the basics of formally working with cell diagrams, and the rest of the paper gives (in my opinion) very compelling motivation to learn this perspective.

As a quick check for comprehension, after digesting some of the Barratt-Jones-Mahowald paper, try to show that the second homotopy group of the mod-2 Moore spectrum contains a nontrivial 2-extension, i.e., is cyclic of order 4. You may find the Toda bracket $$\langle2,\eta,2\rangle$$ useful for this exercise. Appealing to cell diagrams for this problem is extremely illuminating and was the motivating example for me to really learn this perspective.

Considering how many new homotopy theorists try to make a beeline for Higher Algebra, I want to stress that this cell diagram business, though very "classical" in the sense that it was being used 40+ years ago, is still useful for proving new results. One prominent homotopy theorist who uses cell diagrams in his work (and, importantly, papers) is Zhouli Xu. With his collaborators, he's used cell diagrams to improve the 11/8 conjecture result of Furuta, resolve the existence problem for smooth structures on 61-spheres, etc. He has a paper on the Kervaire class $$h_5^2$$ very much inspired by the Barratt-Jones-Mahowald paper that gives further motivation for the cell diagram approach to computations.

While the Higher Algebra approach streamlines constructions in homotopy theory and is certainly worth knowing these days, all the calculations I've ever come across rely on "classical" methods to carry out the main body of the work. If you ever come across a "purely formal" calculation using only the language of higher categories, then there's some classical input being used under the hood, guaranteed. If you really want to understand spectra, rather than the "(adjectives) category" of spectra, you'll need to go old-school.