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

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?

• 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.