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by the discussion here we have for each finite set of primes $\mathfrak{a}$ at least the top part of cohesion in affine $E_\infty$-arithmetic geometry
$(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}}) \colon E_\infty Ring_{nu}^{op}\to E_\infty Ring_{nu}^{op}$
over formal duals of $\mathfrak{a}$-torsion $E_\infty$-rings.
I have a decent geometric intuition of what $\flat_{\mathfrak{a}}$ does, namely the $\mathfrak{a}$-adic completion that it encodes means picking in each $E_\infty$-arithmetic space the collection of all formal neighbourhoods around all its points.
On the other hand, I am presently lacking intuition as to what $\Pi_{\mathfrak{a}}$ is about. Of course the adjoint modality as such tells us that we are to think of this as forming fundamental $\infty$-groupoids/etale homotopy type relative not to points but to formal neighbourhoods. But what I am lacking intuition for presently is why that is given by forming $\mathfrak{a}$-torsion approximation of nonunital $E_\infty$-rings, as it is.
One thought:
if we regard an elliptic curve $C$ as an abelian group hence as a $\mathbb{Z}$-module, then its $p$-torsion approximation is the direct limit overs its groups $C[p^\nu]$ of $p^{\nu}$-torsion points for $\nu$. A $p^\nu$-level structure on $C$ is an isomorphism $\mathbb{Z}/{p^\nu} \times \mathbb{Z}/{p^\nu}\simeq C[p^\nu]$. As $\nu$ tends to infinity, this tends to the actual “geometric realization” in the sense of the fundamental group $\mathbb{Z}\times\mathbb{Z}$ of a complex elliptic curve. Of course in a way this is rather the fundamental group of the dual elliptic curve.
Might this be a hook into conceptually understanding how torsion approximation is analogous to geometric realization?
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