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Notes for the Intro to Stable Homotopy talk in Spring 2021 Juvitop. The talk was given by Dylan Wilson and this post received a lot of help from Niven Achenjang's live-texed notes. I've added some explanations toward the end (especially concerning the construction of Bott elements) that will hopefully help the number theorists in the seminar!
As Dylan mentioned in the talk, a lot of this material can be found in Segal's Categories and Cohomology Theories, which is short and very readable.
We start with an addition exercise. Add these numbers!
you probably did not add them sequentially. You probably grouped them somehow
so really, ever since we learned to add we've understood the notion that the addition properties of an abelian group $A$ can be described by maps
$$ \theta^* : A^n \to A^m\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ (a_1, \ldots, a_n)\mapsto (b_1,\ldots, b_m),\ b_i = \sum_{j\in \theta(i)}a_j $$
indexed by maps $\theta: \{1, 2, \ldots, m\}\to \mathscr{P}(\{1, 2, \ldots, n\})$ where $\mathscr{P}(S)$ indicates the set of subsets of $S$. This description of the composition law of an abelian group is certainly very redundant, but it'll allow us to work up to a definition of something like abelian groups in spaces, these so-called "spectra", which is what we're aiming for today.
We categorized this add-by-grouping procedure using maps of finite sets, so we're probably working with the category of such sets
<aside> ❕ Def. Let $\text{Fin}_*$ denote the category of finite pointed sets, which we can think of as having objects finite sets and morhisms spans $I\hookleftarrow I'\to J$.
</aside>
and functors out of finite sets give composition laws in the following way