Why aren't spectra functions adequate to be the maps in the spectra category?

Question

In his book, "Algebraic Topology - Homotopy and Homology", Switzer define (Definition 8.9) spectra functions as

$f:E\to F $ is a function if is a collection of cellular maps, $\{f_n:n\in\mathbb{Z}\}$ such that $f_{n+1}|_{SE_n}=Sf_{n}$.

In the definition he mentions that we can define compostion of spectra functions in the obvious way. But, right before the definition he says that this notion of function is not adequate to our purpose. Why is that the case? In which way this definition fails to be useful?

Answer 1

If you read the rest of the chapter, you will see that cofinal subspectra play a vital role in the theory of spectra. The goal is to define "maps" between spectra making cofinal subspectra isomorphic to the original spectrum. Functions of spectra are inadequate for this purpose, the inclusion function of a cofinal subspectrum into the original spectrum is no isomorphism. This is the reason why Switzer defines maps of spectra in Definition 8.12.

Answer 2

Let $S^0$ be the sphere spectrum with $S^0_n = S^n$, and let $S^1_n$ be its suspension with $S^1_n = S^{n+1}$. We would like to say that $[S^1, S^0] = \pi_1^{st} = \Bbb Z/2$ with generator given by the stabilization of the Hopf map. But with the naive definition of map of spectra we must start by defining a map $S^1_0 \to S^0_0$ --- that is, a pointed map $S^1 \to S^0$ of spaces --- which is necessarily constant. The map on all higher $S^1_n \to S^0_n$ is an iterated suspension of this, hence also constant. So we don't see the Hopf map in the naive definition.

If you instead define maps on cofinal subspectra you can define your map only starting with the Hopf map $S^1_2 \to S^0_2$.