# Homotopy of Spectra Maps Induced by Homotopy of Functions

So, I'm still working through Adams' lecture notes, and here's something that I haven't been able to immediately suss out:

It is clear that for $F$ a spectrum, $K$ a finite CW-complex and $E$ its associated suspension spectrum, a function $f:\Sigma^nK\to F_n$ induces a map of spectra (since we can just take everything above $\Sigma^nK$ to be the cofinal subspectrum and the suspensions induce all the higher maps).

Now, we are ultimately going to be considering the object $A=\mathrm{colim}_n[\Sigma^nK,F_n]$. So, suppose two maps $f:\Sigma^n\to F_n$ and $g:\Sigma^m\to F_m$ induce the same object in $A$. Then they must (by the nature of the directed system we index our colimit over) coincide at some finite point, i.e. for some $p$ the maps $\Sigma^pK\overset{\Sigma^{p-n}f}\to\Sigma^{p-n}K\to F_p$ and $\Sigma^pK\overset{\Sigma^{p-n}g}\to\Sigma^{p-n}K\to F_p$ are homotopic or define the same object in $[\Sigma^pK,F_p]$.

Adams' next claim is that this homotopy can be extended to the entire spectra (on suitable cofinal subspectra of course) and perhaps it's just my lack of expertise with suspensions and stuff, but I can't quite get it to work out. Something like, if we have a homotopy at a given level then we can just suspend that homotopy to get it on every higher level (i.e. a cofinal subspectrum).

Thanks! -Jon

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If we have a homotopy at one level, that means that there is a map $h:\Sigma^pK\times I\to F_p$. Since this homotopy is pointed, it is constant on the base point, so we might as well consider it as a map $h:\Sigma^pK\wedge I^+\to F_p$. Thus, we have the next map $\Sigma h:\Sigma(\Sigma^pK\wedge I^+)\cong \Sigma^{p+1}K\wedge I^+\to F^{p+1}$ and so on all the way up, which is exactly what we need!