Convenient categories in algebraic topology: their importance, and the role topology plays in their construction

Question

Disclaimer. I have stated three questions but I felt that they are so related that they fit within a single post.

Context. After reading Hatcher's Algebraic Topology I wanted to learn more about homotopy theory and in particular about spectra. I soon found the $n$Lab's Introduction to Stable Homotopy Theory which I'm reading right now. Let me crudely summarize what I've come across.

• We start with the desire to find a category of topological spaces which behaves decently from a category-theoretic viewpoint. The category that we end up finding is the category $\mathbf{Top}^*_{\mathrm{cgwh}}$ of pointed compactly generated weak Hausdorff spaces. The definition is somewhat ad hoc, and there are many other choices besides $\mathbf{Top}^*_{\mathrm{cgwh}}$. This does not concern us.

• We want to endow this with a model structure. We construct the Quillen--Serre structure. It's a rather convoluted struction, and it's not at all the only model structure. Neither of these observations bother us.

• Once we've set this up, we want to consider the category obtained by stabilizing the Quillen adjunction $\Sigma \dashv \Omega$ on $\mathbf{Top}^*_{\mathrm{cgwh}}$, called the stable homotopy category. We end up looking for models of this category, thus arriving at the category of sequential spectra endowed with the stable model structure. The definitions are more convoluted than ever. Still, this leaves us unmoved.

• For some reason the failure of a decent smash product on the category of sequential spectra disappoints us and we end up going even further by considering more highly structured models with yet more complicated definitions and constructions. They go entirely beyond me.

Questions. In all, I am very surprised by what I've read. It has left me with a few general open-ended questions.

• Spectra seemed nice to me because they represent cohomology theories. As such I expected stuff about cohomology. But the emphasis on the notes lies pretty much entirely on finding nice categories. Why do we want them so badly?
• While Hatcher's Algebraic Topology was mostly concerned with investigating spaces, the notes seem to only care for the spaces as being models for the categories we crave for. Does this trend continue as one delves deeper into this subject?
• If the role of spaces has been reduced to models for certain categories, then why do we not think of letting go of them altogether? I mean, let's be real. The journey from your friendly $\mathbf{Top}$ to the daunting $(\mathbf{Top}^*_{\mathrm{cgwh}})_{\rm{Quillen}}$ to the utterly monstrous $\mathrm{SeqSpec}(\mathbf{Top}_{\mathrm{cgwh}})_{\mathrm{stable}}$ (and yet beyond), is to me a horribly unnatural one. It makes me feel that there should be a more royal road to whatever holy grail of categories we are praying for. So are topological spaces even the right framework in the first place? Or am I being ignorant?

Answer 1

This is a pretty open question, so there must be many different perspectives, so let me just offer my opinion.

• At the end of the day, an interesting statement/result/theorem should make sense in the stable homotopy category, which is easier to grasp. But the method/proof (barring categorical generalities) needs access via a category whose homotopy category is spectra. As you've mentioned, there are different choices here. One analogy that some people use is getting a handle on working with abstract vector spaces by choosing a basis. The basis is unimportant, but it makes arguments easier.
• One reason why we need models is to carry out point-set level constructions. This is important when we want to engage spectra seriously. For example, the quest for a good smash product led to a development of highly structured spectra and "spectral algebra", and the model of orthogonal spectra proved useful to study equivariant homotopy theory.
• Topological spaces, as you've seen, can be rather ill-behaved. There is a trend to work instead with simplicial sets, which is more combinatorial in nature and thus less ambiguous. Of course, the two categories are Quillen equivalent (for the Quillen model structure), so their homotopy categories are the same.
• A nice category of spaces (or something like it) is still important in the subject. In some sense, it is the category generated by a point under colimits, and as such is "initial". This universal property (and other characterizations like this) means that spaces will always play a role in algebraic topology.
• If this is your first exposure to spectra, then the construction of a point-set model of spectra might appear a little daunting. But in fact this is a story that is repeated often, and one learns to get used to it (or at least which aspects are important and which are boilerplate).
• Nowadays, an alternative to all these point-set models is provided by the theory of $\infty$-categories, which once set up properly hides all these technical details. For example, we can define the $\infty$-category of spectra as the stable $\infty$-category that is freely generated by one object under colimits, or perhaps as the monoidal unit in the $\infty$-category of stable presentable $\infty$-categories with the Lurie tensor product. But I think the investment to learn enough $\infty$-category theory to set things up this way will be no less than the traditional approach.
• Finally, I should mention that there are parts of stable homotopy theory that can be done without very much regard to foundational matters.