We can view the construction of the real or $p$-adic numbers as the end result of a series of constructions each of which adds more and more structure to the object at hand:
- First, we start with the punctual set $\mathrm{pt}=\{1\}$.
- Then, we consider the free monoid $\mathbb{N}$ on $\mathrm{pt}$, giving us an addition operation $+$ and an additive unit $0$.
- After that, we now add inverses, taking the free group on $\mathbb{N}$ and obtaining the abelian group of integers $\mathbb{Z}$.
- Noticing that $\mathbb{Z}$ also carries a multiplicative monoid structure, we then add inverses again except for $0$, the additive unit, and obtain $\mathbb{Q}$.
- Finally we consider absolute values on $\mathbb{Q}$, and find that besides the trivial absolute value we have the Euclidean absolute value as well as the $p$-adic ones. Metrically completing with respect to these leads one to the real numbers $\mathbb{R}$ or the $p$-adics $\mathbb{Q}_p$.
One thing I've been curious about for the longest time is how far can we carry the steps above when replacing sets by spaces/homotopy types, and thus replacing the punctual set by the punctual space. I think I understand how steps 1–3 should proceed, although I don't know anything about 4–5. Here's how the homotopy version of steps 1–3 would go I think:
This time we start with the punctual space $*$.
Next, we already find "branching answers" like $\mathbb{Q}_p$ and $\mathbb{R}$ in step 5, as we now have to consider the free $\mathbb{E}_k$-monoid on $*$ for $1\leq k\leq\infty$.
In the set theory case the answer is the same for all $k$ (e.g. the free commutative monoid on $\mathrm{pt}$ is also $\mathbb{N}$), although now we have different spaces for each $k$. Indeed, the free $\mathbb{E}_k$-monoid on $*$ is given by $\coprod^{\infty}_{n=0}\mathrm{UConf}_n(\mathbb{R}^k)$, where $\mathrm{UConf}_n(\mathbb{R}^k)$ denotes the unordered configuration spaces of $n$ points in $\mathbb{R}^k$.
The counterpart of step 3 in this setting is also clear: we ought to consider free $\mathbb{E}_k$-groups, which turn out to be $\Omega^kS^k$.
Now $\Omega^kS^k$ also has a multiplicative structure, given by composition of loops instead of concatenation, and I think $\coprod^{\infty}_{n=0}\mathrm{UConf}_n(\mathbb{R}^k)$ also has one, although I'm not sure about the details (the $k=2$ case seems to be related to braid multiplication for instance). That said, I'm not sure how one could proceed as in steps 4–5 with similar universal constructions like rings of fractions and metric completions.
Is there a homotopy-theoretic analogue of these constructions (i.e. rings of fractions, absolute values, and completions for $\mathbb{E}_k$-rings), and if so, what would be the homotopy theoretic counterparts of $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{Q}_p$ in this case?
