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!
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.