The stable homotopy category
A little while ago we discussed the definition of a tensor triangulated category, and in that post we mentioned an example that we didn’t explicitly define, namely the stable homotopy category. The goal for todays post is to fix this. There are many ways of defining it, and some are actually better than others. As the name suggests, the stable homotopy category is a homotopy category, which we have discussed before in the fibration series. But the question is, what is it the homotopy category of? As we remarked in the post on tensor triangulated categories, it is the homotopy category of the category of spectra, and it is here that the different approaches lie. What exactly is the category of spectra, and which spectra are we even talking about? Is it the sequential spectra? or maybe the orthogonal spectra? or perhaps the symmetric ones? maybe $S$-modules or excisive functors? All these names of course means nothing to us yet, as we haven’t properly looked at any of them. We did however meet the $\Omega$-spectrum in an earlier post, but which of the above types does it belong to? ...
